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[Merged by Bors] - chore: remove trailing space in backticks #7617

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2 changes: 1 addition & 1 deletion Mathlib/Algebra/Category/AlgebraCat/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -13,7 +13,7 @@ import Mathlib.Algebra.Category.ModuleCat.Basic
/-!
# Category instance for algebras over a commutative ring

We introduce the bundled category `AlgebraCat` of algebras over a fixed commutative ring `R ` along
We introduce the bundled category `AlgebraCat` of algebras over a fixed commutative ring `R` along
with the forgetful functors to `RingCat` and `ModuleCat`. We furthermore show that the functor
associating to a type the free `R`-algebra on that type is left adjoint to the forgetful functor.
-/
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4 changes: 2 additions & 2 deletions Mathlib/Algebra/Homology/ShortComplex/LeftHomology.lean
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Expand Up @@ -54,11 +54,11 @@ structure LeftHomologyData where
π : K ⟶ H
/-- the kernel condition for `i` -/
wi : i ≫ S.g = 0
/-- `i : K ⟶ S.X₂ ` is a kernel of `g : S.X₂ ⟶ S.X₃` -/
/-- `i : K ⟶ S.X₂` is a kernel of `g : S.X₂ ⟶ S.X₃` -/
hi : IsLimit (KernelFork.ofι i wi)
/-- the cokernel condition for `π` -/
wπ : hi.lift (KernelFork.ofι _ S.zero) ≫ π = 0
/-- `π : K ⟶ H ` is a cokernel of the induced morphism `S.f' : S.X₁ ⟶ K` -/
/-- `π : K ⟶ H` is a cokernel of the induced morphism `S.f' : S.X₁ ⟶ K` -/
hπ : IsColimit (CokernelCofork.ofπ π wπ)

initialize_simps_projections LeftHomologyData (-hi, -hπ)
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2 changes: 1 addition & 1 deletion Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean
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Expand Up @@ -116,7 +116,7 @@ lemma ι_descQ_eq_zero_of_boundary (k : S.X₂ ⟶ A) (x : S.X₃ ⟶ A) (hx : k
congr 1
simp only [← cancel_epi h.p, hx, p_descQ, p_g'_assoc]

/-- For `h : S.RightHomologyData`, this is a restatement of `h.hι `, saying that
/-- For `h : S.RightHomologyData`, this is a restatement of `h.hι`, saying that
`ι : h.H ⟶ h.Q` is a kernel of `h.g' : h.Q ⟶ S.X₃`. -/
def hι' : IsLimit (KernelFork.ofι h.ι h.ι_g') := h.hι

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2 changes: 1 addition & 1 deletion Mathlib/Algebra/Module/Basic.lean
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Expand Up @@ -156,7 +156,7 @@ See note [reducible non-instances]. -/
@[reducible]
def Module.compHom [Semiring S] (f : S →+* R) : Module S M :=
{ MulActionWithZero.compHom M f.toMonoidWithZeroHom, DistribMulAction.compHom M (f : S →* R) with
-- Porting note: the `show f (r + s) • x = f r • x + f s • x ` wasn't needed in mathlib3.
-- Porting note: the `show f (r + s) • x = f r • x + f s • x` wasn't needed in mathlib3.
-- Somehow, now that `SMul` is heterogeneous, it can't unfold earlier fields of a definition for
-- use in later fields. See
-- https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Heterogeneous.20scalar.20multiplication
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2 changes: 1 addition & 1 deletion Mathlib/Algebra/Module/Equiv.lean
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Expand Up @@ -95,7 +95,7 @@ class SemilinearEquivClass (F : Type*) {R S : outParam (Type*)} [Semiring R] [Se
(σ : outParam <| R →+* S) {σ' : outParam <| S →+* R} [RingHomInvPair σ σ'] [RingHomInvPair σ' σ]
(M M₂ : outParam (Type*)) [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module S M₂] extends
AddEquivClass F M M₂ where
/-- Applying a semilinear equivalence `f` over `σ` to `r • x ` equals `σ r • f x`. -/
/-- Applying a semilinear equivalence `f` over `σ` to `r • x` equals `σ r • f x`. -/
map_smulₛₗ : ∀ (f : F) (r : R) (x : M), f (r • x) = σ r • f x
#align semilinear_equiv_class SemilinearEquivClass

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2 changes: 1 addition & 1 deletion Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean
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Expand Up @@ -100,7 +100,7 @@ namespace MorphComponents

variable {X} {n : ℕ} {Z Z' : C} (f : MorphComponents X n Z) (g : X' ⟶ X) (h : Z ⟶ Z')

/-- The morphism `X _[n+1] ⟶ Z ` associated to `f : MorphComponents X n Z`. -/
/-- The morphism `X _[n+1] ⟶ Z` associated to `f : MorphComponents X n Z`. -/
def φ {Z : C} (f : MorphComponents X n Z) : X _[n + 1] ⟶ Z :=
PInfty.f (n + 1) ≫ f.a + ∑ i : Fin (n + 1), (P i).f (n + 1) ≫ X.δ i.rev.succ ≫
f.b (Fin.rev i)
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Original file line number Diff line number Diff line change
Expand Up @@ -38,12 +38,12 @@ theorem atImInfty_mem (S : Set ℍ) : S ∈ atImInfty ↔ ∃ A : ℝ, ∀ z :
simp only [atImInfty_basis.mem_iff, true_and]; rfl
#align upper_half_plane.at_im_infty_mem UpperHalfPlane.atImInfty_mem

/-- A function ` f : ℍ → α` is bounded at infinity if it is bounded along `atImInfty`. -/
/-- A function `f : ℍ → α` is bounded at infinity if it is bounded along `atImInfty`. -/
def IsBoundedAtImInfty {α : Type*} [Norm α] (f : ℍ → α) : Prop :=
BoundedAtFilter atImInfty f
#align upper_half_plane.is_bounded_at_im_infty UpperHalfPlane.IsBoundedAtImInfty

/-- A function ` f : ℍ → α` is zero at infinity it is zero along `atImInfty`. -/
/-- A function `f : ℍ → α` is zero at infinity it is zero along `atImInfty`. -/
def IsZeroAtImInfty {α : Type*} [Zero α] [TopologicalSpace α] (f : ℍ → α) : Prop :=
ZeroAtFilter atImInfty f
#align upper_half_plane.is_zero_at_im_infty UpperHalfPlane.IsZeroAtImInfty
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2 changes: 1 addition & 1 deletion Mathlib/Analysis/Convex/Jensen.lean
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Expand Up @@ -23,7 +23,7 @@ Jensen's inequalities:
* `ConcaveOn.le_map_centerMass`, `ConcaveOn.le_map_sum`: Concave Jensen's inequality.

As corollaries, we get:
* `ConvexOn.exists_ge_of_mem_convexHull `: Maximum principle for convex functions.
* `ConvexOn.exists_ge_of_mem_convexHull`: Maximum principle for convex functions.
* `ConcaveOn.exists_le_of_mem_convexHull`: Minimum principle for concave functions.
-/

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2 changes: 1 addition & 1 deletion Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
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Expand Up @@ -28,7 +28,7 @@ open Finset Set

namespace NNReal

/-- The nonnegative real power function `x^y`, defined for `x : ℝ≥0` and `y : ℝ ` as the
/-- The nonnegative real power function `x^y`, defined for `x : ℝ≥0` and `y : ℝ` as the
restriction of the real power function. For `x > 0`, it is equal to `exp (y log x)`. For `x = 0`,
one sets `0 ^ 0 = 1` and `0 ^ y = 0` for `y ≠ 0`. -/
noncomputable def rpow (x : ℝ≥0) (y : ℝ) : ℝ≥0 :=
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2 changes: 1 addition & 1 deletion Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean
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Expand Up @@ -40,7 +40,7 @@ open scoped Polynomial
for `x ≤ 0`. It is a basic building block to construct smooth partitions of unity. Its main property
is that it vanishes for `x ≤ 0`, it is positive for `x > 0`, and the junction between the two
behaviors is flat enough to retain smoothness. The fact that this function is `C^∞` is proved in
`expNegInvGlue.contDiff `. -/
`expNegInvGlue.contDiff`. -/
def expNegInvGlue (x : ℝ) : ℝ :=
if x ≤ 0 then 0 else exp (-x⁻¹)
#align exp_neg_inv_glue expNegInvGlue
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/ConcreteCategory/Basic.lean
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Expand Up @@ -205,7 +205,7 @@ class HasForget₂ (C : Type u) (D : Type u') [Category.{v} C] [ConcreteCategory
#align category_theory.has_forget₂ CategoryTheory.HasForget₂

/-- The forgetful functor `C ⥤ D` between concrete categories for which we have an instance
`HasForget₂ C `. -/
`HasForget₂ C`. -/
@[reducible]
def forget₂ (C : Type u) (D : Type u') [Category.{v} C] [ConcreteCategory.{w} C]
[Category.{v'} D] [ConcreteCategory.{w} D] [HasForget₂ C D] : C ⥤ D :=
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Endomorphism.lean
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Expand Up @@ -17,7 +17,7 @@ import Mathlib.GroupTheory.GroupAction.Defs
Definition and basic properties of endomorphisms and automorphisms of an object in a category.

For each `X : C`, we provide `CategoryTheory.End X := X ⟶ X` with a monoid structure,
and `CategoryTheory.Aut X := X ≅ X ` with a group structure.
and `CategoryTheory.Aut X := X ≅ X` with a group structure.
-/


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Expand Up @@ -197,7 +197,7 @@ def preservesInitialOfIsIso (f : ⊥_ D ⟶ G.obj (⊥_ C)) [i : IsIso f] :
exact PreservesInitial.ofIsoComparison G
#align category_theory.limits.preserves_initial_of_is_iso CategoryTheory.Limits.preservesInitialOfIsIso

/-- If there is any isomorphism `⊥ ≅ G.obj ⊥ `, then `G` preserves initial objects. -/
/-- If there is any isomorphism `⊥ ≅ G.obj ⊥`, then `G` preserves initial objects. -/
def preservesInitialOfIso (f : ⊥_ D ≅ G.obj (⊥_ C)) : PreservesColimit (Functor.empty C) G :=
preservesInitialOfIsIso G f.hom
#align category_theory.limits.preserves_initial_of_iso CategoryTheory.Limits.preservesInitialOfIso
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
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Expand Up @@ -532,7 +532,7 @@ abbrev prod (X Y : C) [HasBinaryProduct X Y] :=
limit (pair X Y)
#align category_theory.limits.prod CategoryTheory.Limits.prod

/-- If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y ` or
/-- If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y` or
`X ⨿ Y`. -/
abbrev coprod (X Y : C) [HasBinaryCoproduct X Y] :=
colimit (pair X Y)
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1361,7 +1361,7 @@ def getBinaryBiproductData (P Q : C) [HasBinaryBiproduct P Q] : BinaryBiproductD
Classical.choice HasBinaryBiproduct.exists_binary_biproduct
#align category_theory.limits.get_binary_biproduct_data CategoryTheory.Limits.getBinaryBiproductData

/-- A bicone for `P Q ` which is both a limit cone and a colimit cocone. -/
/-- A bicone for `P Q` which is both a limit cone and a colimit cocone. -/
def BinaryBiproduct.bicone (P Q : C) [HasBinaryBiproduct P Q] : BinaryBicone P Q :=
(getBinaryBiproductData P Q).bicone
#align category_theory.limits.binary_biproduct.bicone CategoryTheory.Limits.BinaryBiproduct.bicone
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1329,7 +1329,7 @@ instance pushout.inr_of_epi {X Y Z : C} {f : X ⟶ Y} {g : X ⟶ Z} [HasPushout
PushoutCocone.epi_inr_of_is_pushout_of_epi (colimit.isColimit _)
#align category_theory.limits.pushout.inr_of_epi CategoryTheory.Limits.pushout.inr_of_epi

/-- The map ` X ⨿ Y ⟶ X ⨿[Z] Y` is epi. -/
/-- The map `X ⨿ Y ⟶ X ⨿[Z] Y` is epi. -/
instance epi_coprod_to_pushout {C : Type*} [Category C] {X Y Z : C} (f : X ⟶ Y) (g : X ⟶ Z)
[HasPushout f g] [HasBinaryCoproduct Y Z] :
Epi (coprod.desc pushout.inl pushout.inr : _ ⟶ pushout f g) :=
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2 changes: 1 addition & 1 deletion Mathlib/CategoryTheory/Subobject/Comma.lean
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Expand Up @@ -17,7 +17,7 @@ and `S : D` as a subtype of the subobjects of `A.right`. We deduce that `Structu
well-powered if `C` is.

## Main declarations
* `StructuredArrow.subobjectEquiv `: the order-equivalence between `Subobject A` and a subtype of
* `StructuredArrow.subobjectEquiv`: the order-equivalence between `Subobject A` and a subtype of
`Subobject A.right`.

## Implementation notes
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2 changes: 1 addition & 1 deletion Mathlib/Combinatorics/Catalan.lean
Original file line number Diff line number Diff line change
Expand Up @@ -27,7 +27,7 @@ triangulations of convex polygons.

## Main results

* `catalan_eq_centralBinom_div `: The explicit formula for the Catalan number using the central
* `catalan_eq_centralBinom_div`: The explicit formula for the Catalan number using the central
binomial coefficient, `catalan n = Nat.centralBinom n / (n + 1)`.

* `treesOfNodesEq_card_eq_catalan`: The number of binary trees with `n` internal nodes
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2 changes: 1 addition & 1 deletion Mathlib/Combinatorics/SimpleGraph/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1682,7 +1682,7 @@ abbrev Hom :=
#align simple_graph.hom SimpleGraph.Hom

/-- A graph embedding is an embedding `f` such that for vertices `v w : V`,
`G.Adj (f v) (f w) ↔ G.Adj v w `. Its image is an induced subgraph of G'.
`G.Adj (f v) (f w) ↔ G.Adj v w`. Its image is an induced subgraph of G'.

The notation `G ↪g G'` represents the type of graph embeddings. -/
abbrev Embedding :=
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2 changes: 1 addition & 1 deletion Mathlib/Data/Finsupp/ToDFinsupp.lean
Original file line number Diff line number Diff line change
Expand Up @@ -255,7 +255,7 @@ def finsuppLequivDFinsupp [DecidableEq ι] [Semiring R] [AddCommMonoid M]
map_add' := Finsupp.toDFinsupp_add }
#align finsupp_lequiv_dfinsupp finsuppLequivDFinsupp

-- porting note: `simps` generated as ` ↑(finsuppLequivDFinsupp R).toLinearMap = Finsupp.toDFinsupp`
-- porting note: `simps` generated as `↑(finsuppLequivDFinsupp R).toLinearMap = Finsupp.toDFinsupp`
@[simp]
theorem finsuppLequivDFinsupp_apply_apply [DecidableEq ι] [Semiring R] [AddCommMonoid M]
[∀ m : M, Decidable (m ≠ 0)] [Module R M] :
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2 changes: 1 addition & 1 deletion Mathlib/Data/Seq/Seq.lean
Original file line number Diff line number Diff line change
Expand Up @@ -166,7 +166,7 @@ theorem le_stable (s : Seq α) {m n} (h : m ≤ n) : s.get? m = none → s.get?
exacts [id, fun h2 => al (IH h2)]
#align stream.seq.le_stable Stream'.Seq.le_stable

/-- If a sequence terminated at position `n`, it also terminated at `m ≥ n `. -/
/-- If a sequence terminated at position `n`, it also terminated at `m ≥ n`. -/
theorem terminated_stable : ∀ (s : Seq α) {m n : ℕ}, m ≤ n → s.TerminatedAt m → s.TerminatedAt n :=
le_stable
#align stream.seq.terminated_stable Stream'.Seq.terminated_stable
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2 changes: 1 addition & 1 deletion Mathlib/FieldTheory/Separable.lean
Original file line number Diff line number Diff line change
Expand Up @@ -395,7 +395,7 @@ theorem unique_separable_of_irreducible {f : F[X]} (hf : Irreducible f) (hp : 0

end CharP

/-- If `n ≠ 0` in `F`, then ` X ^ n - a` is separable for any `a ≠ 0`. -/
/-- If `n ≠ 0` in `F`, then `X ^ n - a` is separable for any `a ≠ 0`. -/
theorem separable_X_pow_sub_C {n : ℕ} (a : F) (hn : (n : F) ≠ 0) (ha : a ≠ 0) :
Separable (X ^ n - C a) :=
separable_X_pow_sub_C_unit (Units.mk0 a ha) (IsUnit.mk0 (n : F) hn)
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2 changes: 1 addition & 1 deletion Mathlib/Geometry/Manifold/ContMDiff.lean
Original file line number Diff line number Diff line change
Expand Up @@ -16,7 +16,7 @@ basic properties of these notions.

## Main definitions and statements

Let `M ` and `M'` be two smooth manifolds, with respect to model with corners `I` and `I'`. Let
Let `M` and `M'` be two smooth manifolds, with respect to model with corners `I` and `I'`. Let
`f : M → M'`.

* `ContMDiffWithinAt I I' n f s x` states that the function `f` is `Cⁿ` within the set `s`
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2 changes: 1 addition & 1 deletion Mathlib/GroupTheory/GroupAction/Prod.lean
Original file line number Diff line number Diff line change
Expand Up @@ -13,7 +13,7 @@ import Mathlib.GroupTheory.GroupAction.Defs
This file defines instances for binary product of additive and multiplicative actions and provides
scalar multiplication as a homomorphism from `α × β` to `β`.
## Main declarations
* `smulMulHom `/`smulMonoidHom `: Scalar multiplication bundled as a multiplicative/monoid
* `smulMulHom`/`smulMonoidHom`: Scalar multiplication bundled as a multiplicative/monoid
homomorphism.
## See also
* `Mathlib.GroupTheory.GroupAction.Option`
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2 changes: 1 addition & 1 deletion Mathlib/LinearAlgebra/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1519,7 +1519,7 @@ theorem map_subtype_le (p' : Submodule R p) : map p.subtype p' ≤ p := by
#align submodule.map_subtype_le Submodule.map_subtype_le

/-- Under the canonical linear map from a submodule `p` to the ambient space `M`, the image of the
maximal submodule of `p` is just `p `. -/
maximal submodule of `p` is just `p`. -/
-- @[simp] -- Porting note: simp can prove this
theorem map_subtype_top : map p.subtype (⊤ : Submodule R p) = p := by simp
#align submodule.map_subtype_top Submodule.map_subtype_top
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2 changes: 1 addition & 1 deletion Mathlib/LinearAlgebra/Dimension.lean
Original file line number Diff line number Diff line change
Expand Up @@ -566,7 +566,7 @@ theorem mk_eq_mk_of_basis (v : Basis ι R M) (v' : Basis ι' R M) :
#align mk_eq_mk_of_basis mk_eq_mk_of_basis

/-- Given two bases indexed by `ι` and `ι'` of an `R`-module, where `R` satisfies the invariant
basis number property, an equiv `ι ≃ ι' `. -/
basis number property, an equiv `ι ≃ ι'`. -/
def Basis.indexEquiv (v : Basis ι R M) (v' : Basis ι' R M) : ι ≃ ι' :=
(Cardinal.lift_mk_eq'.1 <| mk_eq_mk_of_basis v v').some
#align basis.index_equiv Basis.indexEquiv
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2 changes: 1 addition & 1 deletion Mathlib/LinearAlgebra/Multilinear/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1276,7 +1276,7 @@ variable (R M M₂)

/-- The space of multilinear maps on `∀ (i : Fin (n+1)), M i` is canonically isomorphic to
the space of linear maps from `M 0` to the space of multilinear maps on
`∀ (i : Fin n), M i.succ `, by separating the first variable. We register this isomorphism as a
`∀ (i : Fin n), M i.succ`, by separating the first variable. We register this isomorphism as a
linear isomorphism in `multilinearCurryLeftEquiv R M M₂`.

The direct and inverse maps are given by `f.uncurryLeft` and `f.curryLeft`. Use these
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2 changes: 1 addition & 1 deletion Mathlib/MeasureTheory/Function/LpSpace.lean
Original file line number Diff line number Diff line change
Expand Up @@ -1082,7 +1082,7 @@ namespace ContinuousLinearMap

variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 F]

/-- Composing `f : Lp ` with `L : E →L[𝕜] F`. -/
/-- Composing `f : Lp` with `L : E →L[𝕜] F`. -/
def compLp (L : E →L[𝕜] F) (f : Lp E p μ) : Lp F p μ :=
L.lipschitz.compLp (map_zero L) f
#align continuous_linear_map.comp_Lp ContinuousLinearMap.compLp
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4 changes: 2 additions & 2 deletions Mathlib/MeasureTheory/Measure/Haar/Quotient.lean
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Expand Up @@ -17,12 +17,12 @@ subgroup of a group `G` on `G` itself.

## Main results

* `MeasureTheory.IsFundamentalDomain.smulInvariantMeasure_map `: given a subgroup `Γ` of a
* `MeasureTheory.IsFundamentalDomain.smulInvariantMeasure_map`: given a subgroup `Γ` of a
topological group `G`, the pushforward to the coset space `G ⧸ Γ` of the restriction of a both
left- and right-invariant measure on `G` to a fundamental domain `𝓕` is a `G`-invariant measure
on `G ⧸ Γ`.

* `MeasureTheory.IsFundamentalDomain.isMulLeftInvariant_map `: given a normal subgroup `Γ` of
* `MeasureTheory.IsFundamentalDomain.isMulLeftInvariant_map`: given a normal subgroup `Γ` of
a topological group `G`, the pushforward to the quotient group `G ⧸ Γ` of the restriction of
a both left- and right-invariant measure on `G` to a fundamental domain `𝓕` is a left-invariant
measure on `G ⧸ Γ`.
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2 changes: 1 addition & 1 deletion Mathlib/NumberTheory/Cyclotomic/Basic.lean
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Expand Up @@ -171,7 +171,7 @@ theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S =
#align is_cyclotomic_extension.subsingleton_iff IsCyclotomicExtension.subsingleton_iff

/-- If `B` is a cyclotomic extension of `A` given by roots of unity of order in `S ∪ T`, then `B`
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 } ` given by
is a cyclotomic extension of `adjoin A { b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1 }` given by
roots of unity of order in `T`. -/
theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by
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2 changes: 1 addition & 1 deletion Mathlib/NumberTheory/NumberField/Units.lean
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Expand Up @@ -181,7 +181,7 @@ distinguished (arbitrary) infinite place, prove that its kernel is the torsion s
follows that `unitLattice` is a free `ℤ`-module (see `unitLattice_moduleFree `) of rank
`card (InfinitePlaces K) - 1` (see `unitLattice_rank`). To prove that the `unitLattice` is a full
`ℤ`-lattice, we need to prove that it is discrete (see `unitLattice_inter_ball_finite`) and that it
spans the full space over `ℝ` (see ` unitLattice_span_eq_top`); this is the main part of the proof,
spans the full space over `ℝ` (see `unitLattice_span_eq_top`); this is the main part of the proof,
see the section `span_top` below for more details.
-/

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2 changes: 1 addition & 1 deletion Mathlib/Order/Filter/Bases.lean
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Expand Up @@ -59,7 +59,7 @@ and consequences are derived.
* `isCountablyGenerated_iff_exists_antitone_basis` : proves a filter is countably generated if and
only if it admits a basis parametrized by a decreasing sequence of sets indexed by `ℕ`.

* `tendsto_iff_seq_tendsto ` : an abstract version of "sequentially continuous implies continuous".
* `tendsto_iff_seq_tendsto` : an abstract version of "sequentially continuous implies continuous".

## Implementation notes

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2 changes: 1 addition & 1 deletion Mathlib/Order/PartialSups.lean
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Expand Up @@ -13,7 +13,7 @@ import Mathlib.Order.ConditionallyCompleteLattice.Finset
# The monotone sequence of partial supremums of a sequence

We define `partialSups : (ℕ → α) → ℕ →o α` inductively. For `f : ℕ → α`, `partialSups f` is
the sequence `f 0 `, `f 0 ⊔ f 1`, `f 0 ⊔ f 1 ⊔ f 2`, ... The point of this definition is that
the sequence `f 0`, `f 0 ⊔ f 1`, `f 0 ⊔ f 1 ⊔ f 2`, ... The point of this definition is that
* it doesn't need a `⨆`, as opposed to `⨆ (i ≤ n), f i` (which also means the wrong thing on
`ConditionallyCompleteLattice`s).
* it doesn't need a `⊥`, as opposed to `(Finset.range (n + 1)).sup f`.
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4 changes: 2 additions & 2 deletions Mathlib/Probability/Independence/Basic.lean
Original file line number Diff line number Diff line change
Expand Up @@ -12,7 +12,7 @@ import Mathlib.Probability.Independence.Kernel

* A family of sets of sets `π : ι → Set (Set Ω)` is independent with respect to a measure `μ` if for
any finite set of indices `s = {i_1, ..., i_n}`, for any sets `f i_1 ∈ π i_1, ..., f i_n ∈ π i_n`,
`μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. It will be used for families of π-systems.
`μ (⋂ i in s, f i) = ∏ i in s, μ (f i)`. It will be used for families of π-systems.
* A family of measurable space structures (i.e. of σ-algebras) is independent with respect to a
measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
define is independent. I.e., `m : ι → MeasurableSpace Ω` is independent with respect to a
Expand Down Expand Up @@ -97,7 +97,7 @@ def IndepSets [MeasurableSpace Ω] (s1 s2 : Set (Set Ω)) (μ : Measure Ω := by
measure `μ` (typically defined on a finer σ-algebra) if the family of sets of measurable sets they
define is independent. `m : ι → MeasurableSpace Ω` is independent with respect to measure `μ` if
for any finite set of indices `s = {i_1, ..., i_n}`, for any sets
`f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i) `. -/
`f i_1 ∈ m i_1, ..., f i_n ∈ m i_n`, then `μ (⋂ i in s, f i) = ∏ i in s, μ (f i)`. -/
def iIndep (m : ι → MeasurableSpace Ω) [MeasurableSpace Ω] (μ : Measure Ω := by volume_tac) :
Prop :=
kernel.iIndep m (kernel.const Unit μ) (Measure.dirac () : Measure Unit)
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