diff --git a/Mathlib/Algebra/Algebra/RestrictScalars.lean b/Mathlib/Algebra/Algebra/RestrictScalars.lean index 0fadc0e547779e..208bca5fdd28a0 100644 --- a/Mathlib/Algebra/Algebra/RestrictScalars.lean +++ b/Mathlib/Algebra/Algebra/RestrictScalars.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Algebra.Tower /-! - # The `RestrictScalars` type alias See the documentation attached to the `RestrictScalars` definition for advice on how and when to diff --git a/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean b/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean index 81af25fb03a6d0..a8f2cf01039adc 100644 --- a/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean +++ b/Mathlib/Algebra/Algebra/Subalgebra/MulOpposite.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Algebra.Subalgebra.Lattice public import Mathlib.Algebra.Ring.Subring.MulOpposite /-! - # Subalgebras of opposite rings For every ring `A` over a commutative ring `R`, we construct an equivalence between diff --git a/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean b/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean index ae303f1a4a8955..80560ba26dd15a 100644 --- a/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean +++ b/Mathlib/Algebra/Algebra/Subalgebra/Rank.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Dimension.Subsingleton public import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition /-! - # Some results on the ranks of subalgebras This file contains some results on the ranks of subalgebras, diff --git a/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean b/Mathlib/Algebra/Category/ContinuousCohomology/Basic.lean new file mode 100644 index 00000000000000..e69de29bb2d1d6 diff --git a/Mathlib/Algebra/Category/Grp/EnoughInjectives.lean b/Mathlib/Algebra/Category/Grp/EnoughInjectives.lean index 4dab687754d73c..69c1f28d2d3177 100644 --- a/Mathlib/Algebra/Category/Grp/EnoughInjectives.lean +++ b/Mathlib/Algebra/Category/Grp/EnoughInjectives.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Category.Grp.EpiMono public import Mathlib.Algebra.Category.Grp.Injective /-! - # Category of abelian groups has enough injectives Given an abelian group `A`, then `i : A ⟶ ∏_{A⋆} ℚ ⧸ ℤ` defined by `i : a ↦ c ↦ c a` is an diff --git a/Mathlib/Algebra/Category/Grp/FiniteGrp.lean b/Mathlib/Algebra/Category/Grp/FiniteGrp.lean index 9773380df2a5fc..27c6785561e663 100644 --- a/Mathlib/Algebra/Category/Grp/FiniteGrp.lean +++ b/Mathlib/Algebra/Category/Grp/FiniteGrp.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Finite.Defs public import Mathlib.Algebra.Category.Grp.Basic /-! - ## Main definitions and results * `FiniteGrp` is the category of finite groups. diff --git a/Mathlib/Algebra/Category/ModuleCat/AB.lean b/Mathlib/Algebra/Category/ModuleCat/AB.lean index 183a3dd6be8be7..dfda5b8bf2827a 100644 --- a/Mathlib/Algebra/Category/ModuleCat/AB.lean +++ b/Mathlib/Algebra/Category/ModuleCat/AB.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Category.ModuleCat.Colimits public import Mathlib.Algebra.Module.Shrink public import Mathlib.CategoryTheory.Abelian.GrothendieckCategory.Basic /-! - # AB axioms in module categories This file proves that the category of modules over a ring satisfies Grothendieck's axioms AB5, AB4, diff --git a/Mathlib/Algebra/Category/ModuleCat/Ext/DimensionShifting.lean b/Mathlib/Algebra/Category/ModuleCat/Ext/DimensionShifting.lean index 6362c36c84d3a1..d2f89b51b4f03b 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Ext/DimensionShifting.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Ext/DimensionShifting.lean @@ -13,7 +13,6 @@ public import Mathlib.Data.Rat.Floor public import Mathlib.Tactic.Continuity /-! - # Dimension Shifting -/ diff --git a/Mathlib/Algebra/Category/ModuleCat/Ext/Finite.lean b/Mathlib/Algebra/Category/ModuleCat/Ext/Finite.lean index d8a073e90cbc36..a47e2a991ee9b8 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Ext/Finite.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Ext/Finite.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat public import Mathlib.RingTheory.Noetherian.Basic /-! - # `Ext`-modules between finitely generated modules over Noetherian rings are finitely generated -/ diff --git a/Mathlib/Algebra/Category/ModuleCat/Ext/HasExt.lean b/Mathlib/Algebra/Category/ModuleCat/Ext/HasExt.lean index e6be84b6c945ca..84eb5a83c8ac42 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Ext/HasExt.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Ext/HasExt.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Category.ModuleCat.Projective public import Mathlib.Algebra.Homology.DerivedCategory.Ext.EnoughProjectives /-! - # HasExt instance for Module Category If we assume `Small.{v} R`, the category `ModuleCat.{v} R` has enough projectives, which allows to diff --git a/Mathlib/Algebra/Category/ModuleCat/Localization.lean b/Mathlib/Algebra/Category/ModuleCat/Localization.lean index 8db1502d63b8e1..7188a54dafc576 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Localization.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Localization.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Module.LocalizedModule.Exact public import Mathlib.RingTheory.Localization.Module /-! - # Localized Module in ModuleCat For a ring `R` satisfying `[Small.{v} R]` and a submonoid `S` of `R`, diff --git a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean index f0044570af4178..1affdbf1ec02bd 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Presheaf.lean @@ -17,7 +17,6 @@ An object `M : PresheafOfModules R` consists of a family of modules of a functorial linear map `M.map f` from `M.obj X` to the restriction of scalars of `M.obj Y` via `R.map f`. - ## Future work * Compare this to the definition as a presheaf of pairs `(R, M)` with specified first part. diff --git a/Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.lean b/Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.lean index 8ceec875a545f7..79f60943d9422d 100644 --- a/Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.lean +++ b/Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Homology.ShortComplex.ModuleCat public import Mathlib.CategoryTheory.Abelian.Projective.Dimension /-! - # Projective Dimension in ModuleCat -/ diff --git a/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean b/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean index 11f3a44f94c3cd..102ead52dea3d3 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Topology/Homology.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Homology.ShortComplex.Abelian public import Mathlib.Algebra.Homology.ShortComplex.PreservesHomology /-! - # `TopModuleCat` is a `CategoryWithHomology` `TopModuleCat R`, the category of topological `R`-modules, is not an abelian category. diff --git a/Mathlib/Algebra/Category/ModuleCat/Ulift.lean b/Mathlib/Algebra/Category/ModuleCat/Ulift.lean index d8ede937f0c089..6cf64f65dec0dc 100644 --- a/Mathlib/Algebra/Category/ModuleCat/Ulift.lean +++ b/Mathlib/Algebra/Category/ModuleCat/Ulift.lean @@ -13,7 +13,6 @@ public import Mathlib.CategoryTheory.Preadditive.Injective.Preserves public import Mathlib.CategoryTheory.Preadditive.Projective.Preserves /-! - # Ulift functor for ModuleCat In this file, we define the obvious functor `ModuleCat.{v} R ⥤ ModuleCat.{max v v'} R` and prove diff --git a/Mathlib/Algebra/Category/Ring/FinitePresentation.lean b/Mathlib/Algebra/Category/Ring/FinitePresentation.lean index a6af210fb9b3b5..41ecd4c325775d 100644 --- a/Mathlib/Algebra/Category/Ring/FinitePresentation.lean +++ b/Mathlib/Algebra/Category/Ring/FinitePresentation.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.EssentialFiniteness public import Mathlib.RingTheory.FinitePresentation /-! - # Finitely presentable objects in `Under R` with `R : CommRingCat` In this file, we show that finitely presented algebras are finitely presentable in `Under R`, diff --git a/Mathlib/Algebra/Central/TensorProduct.lean b/Mathlib/Algebra/Central/TensorProduct.lean index 1a73a172e2199c..46cc343e9a423c 100644 --- a/Mathlib/Algebra/Central/TensorProduct.lean +++ b/Mathlib/Algebra/Central/TensorProduct.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Flat.Basic public import Mathlib.LinearAlgebra.Basis.VectorSpace /-! - # Lemmas about tensor products of central algebras In this file we prove for algebras `B` and `C` over a field `K` that if `B ⊗[K] C` is a central diff --git a/Mathlib/Algebra/CharP/Algebra.lean b/Mathlib/Algebra/CharP/Algebra.lean index c3fdd42442d0d7..da5f46097ee633 100644 --- a/Mathlib/Algebra/CharP/Algebra.lean +++ b/Mathlib/Algebra/CharP/Algebra.lean @@ -18,7 +18,6 @@ In this file we describe the characteristic of `R`-algebras. In particular we are interested in the characteristic of free algebras over `R` and the fraction field `FractionRing R`. - ## Main results - `charP_of_injective_algebraMap` If `R →+* A` is an injective algebra map diff --git a/Mathlib/Algebra/CharP/IntermediateField.lean b/Mathlib/Algebra/CharP/IntermediateField.lean index d8aa777908292f..87435094704bf4 100644 --- a/Mathlib/Algebra/CharP/IntermediateField.lean +++ b/Mathlib/Algebra/CharP/IntermediateField.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.CharP.Algebra public import Mathlib.FieldTheory.IntermediateField.Basic /-! - # Characteristic of intermediate fields This file contains some convenient instances for determining the characteristic of diff --git a/Mathlib/Algebra/CharZero/Defs.lean b/Mathlib/Algebra/CharZero/Defs.lean index 13ab36b10a6970..eece1507a35f82 100644 --- a/Mathlib/Algebra/CharZero/Defs.lean +++ b/Mathlib/Algebra/CharZero/Defs.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Int.Cast.Defs public import Mathlib.Logic.Basic /-! - # Characteristic zero A ring `R` is called of characteristic zero if every natural number `n` is non-zero when considered diff --git a/Mathlib/Algebra/DirectSum/Decomposition.lean b/Mathlib/Algebra/DirectSum/Decomposition.lean index 32e9581ec7509f..048b5f31bfa012 100644 --- a/Mathlib/Algebra/DirectSum/Decomposition.lean +++ b/Mathlib/Algebra/DirectSum/Decomposition.lean @@ -17,7 +17,6 @@ public import Mathlib.Algebra.Module.Submodule.Basic an additive monoid `M` into a family of additive submonoids `ℳ` * `DirectSum.decompose ℳ`: The canonical equivalence provided by the above typeclass - ## Main statements * `DirectSum.Decomposition.isInternal`: The link to `DirectSum.IsInternal`. diff --git a/Mathlib/Algebra/EuclideanDomain/Defs.lean b/Mathlib/Algebra/EuclideanDomain/Defs.lean index ab7635c4a5e85b..702d424fa152f2 100644 --- a/Mathlib/Algebra/EuclideanDomain/Defs.lean +++ b/Mathlib/Algebra/EuclideanDomain/Defs.lean @@ -55,7 +55,6 @@ value of `j`. [MR399081] * [M. Nagata, *On Euclid algorithm*][MR541021] - ## Tags Euclidean domain, transfinite Euclidean domain, Bézout's lemma diff --git a/Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean b/Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean index d22d057cea4e58..b808fd60cd7d98 100644 --- a/Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean +++ b/Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed public import Mathlib.RingTheory.Polynomial.Eisenstein.Basic /-! - # GCD domains are integrally closed -/ diff --git a/Mathlib/Algebra/Group/Indicator.lean b/Mathlib/Algebra/Group/Indicator.lean index 9466e061e14c3a..fbb6e8364062b4 100644 --- a/Mathlib/Algebra/Group/Indicator.lean +++ b/Mathlib/Algebra/Group/Indicator.lean @@ -18,7 +18,6 @@ In this file, we prove basic results about the indicator of a set. - `Set.indicator (s : Set α) (f : α → β) (a : α)` is `f a` if `a ∈ s` and is `0` otherwise. - `Set.mulIndicator (s : Set α) (f : α → β) (a : α)` is `f a` if `a ∈ s` and is `1` otherwise. - ## Implementation note In mathematics, an indicator function or a characteristic function is a function diff --git a/Mathlib/Algebra/Group/Submonoid/Units.lean b/Mathlib/Algebra/Group/Submonoid/Units.lean index 43f53ae8b81f86..d430e5bedf6e6a 100644 --- a/Mathlib/Algebra/Group/Submonoid/Units.lean +++ b/Mathlib/Algebra/Group/Submonoid/Units.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Group.Submonoid.Pointwise public import Mathlib.Algebra.Group.Subgroup.Lattice /-! - # Submonoid of units Given a submonoid `S` of a monoid `M`, we define the subgroup `S.units` as the units of `S` as a diff --git a/Mathlib/Algebra/GroupWithZero/Hom.lean b/Mathlib/Algebra/GroupWithZero/Hom.lean index 74531ee178a91f..b6a8a11bf77d30 100644 --- a/Mathlib/Algebra/GroupWithZero/Hom.lean +++ b/Mathlib/Algebra/GroupWithZero/Hom.lean @@ -16,7 +16,6 @@ This file defines homomorphisms of monoids with zero. We also define coercion to a function, and usual operations: composition, identity homomorphism, pointwise multiplication and pointwise inversion. - ## Notation * `→*₀`: `MonoidWithZeroHom`, the type of bundled `MonoidWithZero` homs. Also use for diff --git a/Mathlib/Algebra/Homology/DerivedCategory/Ext/MapBijective.lean b/Mathlib/Algebra/Homology/DerivedCategory/Ext/MapBijective.lean index ac709e0e840fc5..f244075d8c27fa 100644 --- a/Mathlib/Algebra/Homology/DerivedCategory/Ext/MapBijective.lean +++ b/Mathlib/Algebra/Homology/DerivedCategory/Ext/MapBijective.lean @@ -14,7 +14,6 @@ public import Mathlib.CategoryTheory.Preadditive.Injective.Preserves public import Mathlib.CategoryTheory.Preadditive.Projective.Preserves /-! - # Bijections Between Ext In this file, we show that the maps between `Ext` induced diff --git a/Mathlib/Algebra/Lie/Derivation/BaseChange.lean b/Mathlib/Algebra/Lie/Derivation/BaseChange.lean index cf97e571ae747d..0ca9ea607a87b4 100644 --- a/Mathlib/Algebra/Lie/Derivation/BaseChange.lean +++ b/Mathlib/Algebra/Lie/Derivation/BaseChange.lean @@ -16,7 +16,6 @@ When, given an `R`-algebra `A` and an `R`-Lie algebra `L` the (Lie algebra) base both derivations of `A` and Lie derivations of `L` induce Lie derivations of `A ⊗[R] L`. Moreover, both these procedures are Lie algebra homomorphisms themselves. - ## Tags lie algebra, extension of scalars, base change, derivation diff --git a/Mathlib/Algebra/Lie/SemiDirect.lean b/Mathlib/Algebra/Lie/SemiDirect.lean index 1aadbffaf653cd..5f14e97da22e83 100644 --- a/Mathlib/Algebra/Lie/SemiDirect.lean +++ b/Mathlib/Algebra/Lie/SemiDirect.lean @@ -19,7 +19,6 @@ algebra homomorphism `ψ : L → LieDerivation R K K`, the underlying set of t itself a Lie algebra and that it fits into an exact sequence `H → (SemiDirectSum K L ψ) → L`, i.e. forms an extension of `L`. - ## References * https://en.wikipedia.org/wiki/Lie_algebra_extension#By_semidirect_sum diff --git a/Mathlib/Algebra/Lie/Sl2.lean b/Mathlib/Algebra/Lie/Sl2.lean index 56e7839a424f6b..c96399b8d55bde 100644 --- a/Mathlib/Algebra/Lie/Sl2.lean +++ b/Mathlib/Algebra/Lie/Sl2.lean @@ -11,7 +11,6 @@ public import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition public import Mathlib.Algebra.Lie.Weights.Basic /-! - # The Lie algebra `sl₂` and its representations The Lie algebra `sl₂` is the unique simple Lie algebra of minimal rank, 1, and as such occupies a diff --git a/Mathlib/Algebra/Module/FinitePresentation.lean b/Mathlib/Algebra/Module/FinitePresentation.lean index de0440401bcbe2..915f48251aceb1 100644 --- a/Mathlib/Algebra/Module/FinitePresentation.lean +++ b/Mathlib/Algebra/Module/FinitePresentation.lean @@ -14,7 +14,6 @@ public import Mathlib.RingTheory.Noetherian.Basic public import Mathlib.RingTheory.TensorProduct.Finite /-! - # Finitely Presented Modules ## Main definition @@ -37,7 +36,6 @@ Suppose `0 → K → M → N → 0` is an exact sequence of `R`-modules. - `Module.FinitePresentation.isLocalizedModule_map`: If `M` and `N` are `R`-modules and `M` is f.p., and `S` is a submonoid of `R`, then `Hom(Mₛ, Nₛ)` is the localization of `Hom(M, N)`. - Also the instances finite + free => f.p. => finite are also provided ## TODO @@ -50,7 +48,6 @@ Suppose `S` is an `R`-algebra, `M` is an `S`-module. Then In particular, 4. `S` is f.p. as an `R`-module iff it is f.p. as an algebra and is finite as a module. - For finitely presented algebras, see `Algebra.FinitePresentation` in file `Mathlib/RingTheory/FinitePresentation.lean`. -/ diff --git a/Mathlib/Algebra/Module/LocalizedModule/Int.lean b/Mathlib/Algebra/Module/LocalizedModule/Int.lean index e43294f55d06fe..da4c911eea140c 100644 --- a/Mathlib/Algebra/Module/LocalizedModule/Int.lean +++ b/Mathlib/Algebra/Module/LocalizedModule/Int.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Module.LocalizedModule.Basic /-! - # Integer elements of a localized module This is a mirror of the corresponding notion for localizations of rings. diff --git a/Mathlib/Algebra/Module/Projective.lean b/Mathlib/Algebra/Module/Projective.lean index c6bd0bb04be016..d6e5ab0fb13827 100644 --- a/Mathlib/Algebra/Module/Projective.lean +++ b/Mathlib/Algebra/Module/Projective.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.TensorProduct.Basis public import Mathlib.Logic.UnivLE /-! - # Projective modules This file contains a definition of a projective module, the proof that diff --git a/Mathlib/Algebra/Module/SnakeLemma.lean b/Mathlib/Algebra/Module/SnakeLemma.lean index 7c694eb66871e1..d3d14744fab8f2 100644 --- a/Mathlib/Algebra/Module/SnakeLemma.lean +++ b/Mathlib/Algebra/Module/SnakeLemma.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Exact.Basic /-! - # The snake lemma in terms of modules The snake lemma is proven in `Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean` for all abelian diff --git a/Mathlib/Algebra/Module/Submodule/Defs.lean b/Mathlib/Algebra/Module/Submodule/Defs.lean index 9b7075715d3ddc..802b2f4ea2c582 100644 --- a/Mathlib/Algebra/Module/Submodule/Defs.lean +++ b/Mathlib/Algebra/Module/Submodule/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.GroupAction.SubMulAction public import Mathlib.Algebra.Group.Submonoid.Basic /-! - # Submodules of a module In this file we define diff --git a/Mathlib/Algebra/Module/Submodule/IterateMapComap.lean b/Mathlib/Algebra/Module/Submodule/IterateMapComap.lean index 669f714ed2679d..ddd27d4a4533d1 100644 --- a/Mathlib/Algebra/Module/Submodule/IterateMapComap.lean +++ b/Mathlib/Algebra/Module/Submodule/IterateMapComap.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Module.Submodule.Ker /-! - # Iterate maps and comaps of submodules Some preliminary work for establishing the strong rank condition for Noetherian rings. diff --git a/Mathlib/Algebra/Module/Submodule/Lattice.lean b/Mathlib/Algebra/Module/Submodule/Lattice.lean index 8f2375d8309044..82f90739a305c4 100644 --- a/Mathlib/Algebra/Module/Submodule/Lattice.lean +++ b/Mathlib/Algebra/Module/Submodule/Lattice.lean @@ -22,7 +22,6 @@ This file defines the lattice structure on submodules, `Submodule.CompleteLattic defined as `{0}` and `⊓` defined as intersection of the underlying carrier. If `p` and `q` are submodules of a module, `p ≤ q` means that `p ⊆ q`. - ## Implementation notes This structure should match the `AddSubmonoid.CompleteLattice` structure, and we should try diff --git a/Mathlib/Algebra/Module/Submodule/LinearMap.lean b/Mathlib/Algebra/Module/Submodule/LinearMap.lean index 928792321523ec..2fc6dd74f4c00c 100644 --- a/Mathlib/Algebra/Module/Submodule/LinearMap.lean +++ b/Mathlib/Algebra/Module/Submodule/LinearMap.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Module.Submodule.Defs public import Mathlib.Algebra.BigOperators.Group.Finset.Defs /-! - # Linear maps involving submodules of a module In this file we define a number of linear maps involving submodules of a module. diff --git a/Mathlib/Algebra/Module/Submodule/RestrictScalars.lean b/Mathlib/Algebra/Module/Submodule/RestrictScalars.lean index a1d249200d4eee..3d811bf16b59b8 100644 --- a/Mathlib/Algebra/Module/Submodule/RestrictScalars.lean +++ b/Mathlib/Algebra/Module/Submodule/RestrictScalars.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Module.Submodule.Lattice public import Mathlib.Order.Hom.CompleteLattice /-! - # Restriction of scalars for submodules If semiring `S` acts on a semiring `R` and `M` is a module over both (compatibly with this action) diff --git a/Mathlib/Algebra/MonoidAlgebra/Degree.lean b/Mathlib/Algebra/MonoidAlgebra/Degree.lean index 259226a19b845f..93f93940cfdb86 100644 --- a/Mathlib/Algebra/MonoidAlgebra/Degree.lean +++ b/Mathlib/Algebra/MonoidAlgebra/Degree.lean @@ -27,7 +27,6 @@ variable {R R' A T B ι : Type*} namespace AddMonoidAlgebra /-! - ## sup-degree and inf-degree of an `AddMonoidAlgebra` Let `R` be a semiring and let `A` be a `SemilatticeSup`. @@ -78,7 +77,6 @@ variable [Semiring R] section ExplicitDegrees /-! - In this section, we use `degb` and `degt` to denote "degree functions" on `A` with values in a type with *b*ot or *t*op respectively. -/ diff --git a/Mathlib/Algebra/MvPolynomial/Monad.lean b/Mathlib/Algebra/MvPolynomial/Monad.lean index 7dc05a1dfde106..10ba4b2d94f2af 100644 --- a/Mathlib/Algebra/MvPolynomial/Monad.lean +++ b/Mathlib/Algebra/MvPolynomial/Monad.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.MvPolynomial.Rename public import Mathlib.Algebra.MvPolynomial.Variables /-! - # Monad operations on `MvPolynomial` This file defines two monadic operations on `MvPolynomial`. Given `p : MvPolynomial σ R`, diff --git a/Mathlib/Algebra/MvPolynomial/Variables.lean b/Mathlib/Algebra/MvPolynomial/Variables.lean index a832754446ea5a..1abd57ff89bccf 100644 --- a/Mathlib/Algebra/MvPolynomial/Variables.lean +++ b/Mathlib/Algebra/MvPolynomial/Variables.lean @@ -16,7 +16,6 @@ This file establishes many results about the variable sets of a multivariate pol The *variable set* of a polynomial $P \in R[X]$ is a `Finset` containing each $x \in X$ that appears in a monomial in $P$. - ## Main declarations * `MvPolynomial.vars p` : the finset of variables occurring in `p`. diff --git a/Mathlib/Algebra/Order/Antidiag/FinsuppEquiv.lean b/Mathlib/Algebra/Order/Antidiag/FinsuppEquiv.lean index 46e04d0d7535ae..8aac856259e8eb 100644 --- a/Mathlib/Algebra/Order/Antidiag/FinsuppEquiv.lean +++ b/Mathlib/Algebra/Order/Antidiag/FinsuppEquiv.lean @@ -11,7 +11,6 @@ public import Mathlib.Data.Finsupp.Multiset import Mathlib.Data.Sym.Card /-! - # Equivalence between `Finset.finsuppAntidiag` and `Sym` This file collects further results about equivalence and cardinality related to diff --git a/Mathlib/Algebra/Order/Chebyshev.lean b/Mathlib/Algebra/Order/Chebyshev.lean index e450d2b7f6c12d..d1f14cc3ed48be 100644 --- a/Mathlib/Algebra/Order/Chebyshev.lean +++ b/Mathlib/Algebra/Order/Chebyshev.lean @@ -19,7 +19,6 @@ This file proves the Chebyshev sum inequality. Chebyshev's inequality states `(∑ i ∈ s, f i) * (∑ i ∈ s, g i) ≤ #s * ∑ i ∈ s, f i * g i` when `f g : ι → α` monovary, and the reverse inequality when `f` and `g` antivary. - ## Main declarations * `MonovaryOn.sum_mul_sum_le_card_mul_sum`: Chebyshev's inequality. diff --git a/Mathlib/Algebra/Order/GroupWithZero/WithZero.lean b/Mathlib/Algebra/Order/GroupWithZero/WithZero.lean index 1835390dcaf2b0..f620504ba1b592 100644 --- a/Mathlib/Algebra/Order/GroupWithZero/WithZero.lean +++ b/Mathlib/Algebra/Order/GroupWithZero/WithZero.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Order.GroupWithZero.Basic public import Mathlib.Algebra.Order.GroupWithZero.Canonical /-! - # Covariant instances on `WithZero` Adding a zero to a type with a preorder and multiplication which satisfies some diff --git a/Mathlib/Algebra/Order/Hom/TypeTags.lean b/Mathlib/Algebra/Order/Hom/TypeTags.lean index bb92992df9519e..7804ef1c4c6919 100644 --- a/Mathlib/Algebra/Order/Hom/TypeTags.lean +++ b/Mathlib/Algebra/Order/Hom/TypeTags.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Order.Hom.Monoid public import Mathlib.Algebra.Order.Monoid.Unbundled.TypeTags /-! - # Order Monoid Isomorphisms on `Additive` and `Multiplicative`. -/ diff --git a/Mathlib/Algebra/Order/Module/Defs.lean b/Mathlib/Algebra/Order/Module/Defs.lean index 88019b5c7d3466..e678b2d4f956b9 100644 --- a/Mathlib/Algebra/Order/Module/Defs.lean +++ b/Mathlib/Algebra/Order/Module/Defs.lean @@ -29,7 +29,6 @@ purposes, and the system is set up so that they imply the correct granular typec If those are enough for you, you may stop reading here! Else, beware that what follows is a bit technical. - In all that follows, `α` and `β` are orders which have a `0` and such that `α` acts on `β` by scalar multiplication. Note however that we do not use lawfulness of this action in most of the file. Hence `•` should be considered here as a mostly arbitrary function `α → β → β`. diff --git a/Mathlib/Algebra/Order/Module/Synonym.lean b/Mathlib/Algebra/Order/Module/Synonym.lean index cb86b6a1f2cad1..41a9f8a1900671 100644 --- a/Mathlib/Algebra/Order/Module/Synonym.lean +++ b/Mathlib/Algebra/Order/Module/Synonym.lean @@ -12,7 +12,6 @@ public import Mathlib.Algebra.Order.Ring.Synonym /-! # Action instances for `OrderDual` - This PR transfers group action with zero instances from a type `α` to `αᵒᵈ` and `Lex α`. Note that the `SMul` instances are already defined in `Mathlib/Algebra/Order/Group/Synonym.lean`. diff --git a/Mathlib/Algebra/Order/Monoid/LocallyFiniteOrder.lean b/Mathlib/Algebra/Order/Monoid/LocallyFiniteOrder.lean index 0675be29429bd3..5034c1c8c39008 100644 --- a/Mathlib/Algebra/Order/Monoid/LocallyFiniteOrder.lean +++ b/Mathlib/Algebra/Order/Monoid/LocallyFiniteOrder.lean @@ -16,7 +16,6 @@ public import Mathlib.Algebra.Group.Embedding public import Mathlib.Order.Interval.Finset.Basic /-! - # Locally Finite Linearly Ordered Abelian Groups ## Main results diff --git a/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean b/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean index 0e4a1637743fee..83b4b68342e911 100644 --- a/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean +++ b/Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.Order.Basic public import Mathlib.Order.Monotone.Defs /-! - # Covariants and contravariants This file contains general lemmas and instances to work with the interactions between a relation and diff --git a/Mathlib/Algebra/Polynomial/Basis.lean b/Mathlib/Algebra/Polynomial/Basis.lean index c0028424bcd382..408ac75d7a1d08 100644 --- a/Mathlib/Algebra/Polynomial/Basis.lean +++ b/Mathlib/Algebra/Polynomial/Basis.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Polynomial.Basic public import Mathlib.LinearAlgebra.Basis.Defs /-! - # Basis of a polynomial ring -/ diff --git a/Mathlib/Algebra/Polynomial/PartialFractions.lean b/Mathlib/Algebra/Polynomial/PartialFractions.lean index 54231e42ec29a6..9833d1a59e950b 100644 --- a/Mathlib/Algebra/Polynomial/PartialFractions.lean +++ b/Mathlib/Algebra/Polynomial/PartialFractions.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Polynomial.Div public import Mathlib.RingTheory.Coprime.Lemmas /-! - # Partial fractions For `f, g : R[X]`, if `g` is expressed as a product `g₁ ^ n₁ * g₂ ^ n₂ * ... * gₙ ^ nₙ`, @@ -23,11 +22,9 @@ Since polynomials do not have a division, the main theorem `mul_prod_pow_inverse_eq_quo_add_sum_rem_mul_pow_inverse` is stated in an `R[X]`-algebra `K` containing inverses `giᵢ` for each polynomial `gᵢ` occurring in the denominator. - These results were formalised by the Xena Project, at the suggestion of Patrick Massot. - ## Main results * `mul_prod_pow_inverse_eq_quo_add_sum_rem_mul_pow_inverse`: Partial fraction decomposition for diff --git a/Mathlib/Algebra/Polynomial/RuleOfSigns.lean b/Mathlib/Algebra/Polynomial/RuleOfSigns.lean index 7cddfd744d4758..8773cc0c9df92b 100644 --- a/Mathlib/Algebra/Polynomial/RuleOfSigns.lean +++ b/Mathlib/Algebra/Polynomial/RuleOfSigns.lean @@ -12,7 +12,6 @@ public import Mathlib.Data.List.Destutter public import Mathlib.Data.Sign.Basic /-! - # Descartes' Rule of Signs We define the "sign changes" in the coefficients of a polynomial, and prove Descartes' diff --git a/Mathlib/Algebra/Polynomial/Sequence.lean b/Mathlib/Algebra/Polynomial/Sequence.lean index fa7c4361348825..998284cb039352 100644 --- a/Mathlib/Algebra/Polynomial/Sequence.lean +++ b/Mathlib/Algebra/Polynomial/Sequence.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Basis.Basic public import Mathlib.RingTheory.Polynomial.Basic /-! - # Polynomial sequences We define polynomial sequences – sequences of polynomials `a₀, a₁, ...` such that the polynomial diff --git a/Mathlib/Algebra/QuadraticAlgebra/Defs.lean b/Mathlib/Algebra/QuadraticAlgebra/Defs.lean index dbe64d0bfbd622..72d83a7c4cfdc8 100644 --- a/Mathlib/Algebra/QuadraticAlgebra/Defs.lean +++ b/Mathlib/Algebra/QuadraticAlgebra/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.FreeModule.Finite.Basic /-! - # Quadratic Algebra In this file we define the quadratic algebra `QuadraticAlgebra R a b` over a commutative ring `R`, diff --git a/Mathlib/Algebra/Ring/Subring/MulOpposite.lean b/Mathlib/Algebra/Ring/Subring/MulOpposite.lean index f92b3d0ab594db..5db129dedafabb 100644 --- a/Mathlib/Algebra/Ring/Subring/MulOpposite.lean +++ b/Mathlib/Algebra/Ring/Subring/MulOpposite.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Ring.Subsemiring.MulOpposite public import Mathlib.Algebra.Ring.Subring.Basic /-! - # Subring of opposite rings For every ring `R`, we construct an equivalence between subrings of `R` and that of `Rᵐᵒᵖ`. diff --git a/Mathlib/Algebra/Ring/Subring/Units.lean b/Mathlib/Algebra/Ring/Subring/Units.lean index 0a1372cd5cc622..c768a43ab56bf5 100644 --- a/Mathlib/Algebra/Ring/Subring/Units.lean +++ b/Mathlib/Algebra/Ring/Subring/Units.lean @@ -14,7 +14,6 @@ public import Mathlib.Algebra.Ring.Subring.Basic import Mathlib.Algebra.Group.Submonoid.Units /-! - # Unit subgroups of a ring -/ diff --git a/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean b/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean index a2e677732e4c03..0c00d5cd27ef42 100644 --- a/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean +++ b/Mathlib/Algebra/Ring/Subsemiring/MulOpposite.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Ring.Subsemiring.Basic public import Mathlib.Algebra.Ring.Opposite /-! - # Subsemiring of opposite semirings For every semiring `R`, we construct an equivalence between subsemirings of `R` and that of `Rᵐᵒᵖ`. diff --git a/Mathlib/Algebra/Tropical/Basic.lean b/Mathlib/Algebra/Tropical/Basic.lean index e79672ef90f364..d307ced8bd3556 100644 --- a/Mathlib/Algebra/Tropical/Basic.lean +++ b/Mathlib/Algebra/Tropical/Basic.lean @@ -13,7 +13,6 @@ public import Mathlib.Algebra.Ring.Defs public import Mathlib.Order.Hom.Basic /-! - # Tropical algebraic structures This file defines algebraic structures of the (min-)tropical numbers, up to the tropical semiring. diff --git a/Mathlib/Algebra/Tropical/BigOperators.lean b/Mathlib/Algebra/Tropical/BigOperators.lean index c9dd058dada748..37418254664161 100644 --- a/Mathlib/Algebra/Tropical/BigOperators.lean +++ b/Mathlib/Algebra/Tropical/BigOperators.lean @@ -11,7 +11,6 @@ public import Mathlib.Order.ConditionallyCompleteLattice.Finset public import Mathlib.Algebra.BigOperators.Group.Finset.Basic /-! - # Tropicalization of finitary operations This file provides the "big-op" or notation-based finitary operations on tropicalized types. diff --git a/Mathlib/Algebra/Tropical/Lattice.lean b/Mathlib/Algebra/Tropical/Lattice.lean index 553d83ee183b69..81c8c67f6b1f96 100644 --- a/Mathlib/Algebra/Tropical/Lattice.lean +++ b/Mathlib/Algebra/Tropical/Lattice.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Tropical.Basic public import Mathlib.Order.ConditionallyCompleteLattice.Basic /-! - # Order on tropical algebraic structure This file defines the orders induced on tropical algebraic structures by the underlying type. diff --git a/Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean b/Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean index 0bfee543aed3b0..73cae0b06e064a 100644 --- a/Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean +++ b/Mathlib/AlgebraicGeometry/AffineTransitionLimit.lean @@ -15,7 +15,6 @@ public import Mathlib.CategoryTheory.Limits.Types.ColimitTypeFiltered public import Mathlib.CategoryTheory.Monad.Limits /-! - # Inverse limits of schemes with affine transition maps In this file, we develop API for inverse limits of schemes with affine transition maps, @@ -372,7 +371,6 @@ lemma Scheme.compactSpace_of_isLimit [IsCofiltered I] exact QuasiCompact.compactSpace_of_compactSpace (c.π.app i) /-! - ## Cofiltered Limits and Schemes of Finite Type Given a cofiltered diagram `D` of quasi-compact `S`-schemes with affine transition maps, diff --git a/Mathlib/AlgebraicGeometry/Birational/RationalMap.lean b/Mathlib/AlgebraicGeometry/Birational/RationalMap.lean index 4a0717389ddacb..12b2a587ee8aad 100644 --- a/Mathlib/AlgebraicGeometry/Birational/RationalMap.lean +++ b/Mathlib/AlgebraicGeometry/Birational/RationalMap.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.SpreadingOut public import Mathlib.AlgebraicGeometry.FunctionField public import Mathlib.AlgebraicGeometry.Morphisms.Separated /-! - # Rational maps between schemes ## Main definitions diff --git a/Mathlib/AlgebraicGeometry/ColimitsOver.lean b/Mathlib/AlgebraicGeometry/ColimitsOver.lean index 583a137e8bbe08..25d28b3b30132f 100644 --- a/Mathlib/AlgebraicGeometry/ColimitsOver.lean +++ b/Mathlib/AlgebraicGeometry/ColimitsOver.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.RelativeGluing public import Mathlib.CategoryTheory.MorphismProperty.OverAdjunction /-! - # Colimits in `P.Over ⊤ S` Let `P` be a morphism property in the category of schemes and `S` be a scheme. Let diff --git a/Mathlib/AlgebraicGeometry/Cover/Over.lean b/Mathlib/AlgebraicGeometry/Cover/Over.lean index 6c01bf3df1f520..531a9a6269306a 100644 --- a/Mathlib/AlgebraicGeometry/Cover/Over.lean +++ b/Mathlib/AlgebraicGeometry/Cover/Over.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap public import Mathlib.CategoryTheory.Limits.MorphismProperty /-! - # Covers of schemes over a base In this file we define the typeclass `Cover.Over`. For a cover `𝒰` of an `S`-scheme `X`, diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean index f183afc8eca54e..d1c7c31e73f6c0 100644 --- a/Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean +++ b/Mathlib/AlgebraicGeometry/EllipticCurve/IsomOfJ.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.EllipticCurve.NormalForms public import Mathlib.FieldTheory.IsSepClosed /-! - # Elliptic curves with same j-invariants are isomorphic ## Main results diff --git a/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean b/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean index 294a9ebfcb05d9..9803c04e5455c5 100644 --- a/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean +++ b/Mathlib/AlgebraicGeometry/EllipticCurve/NormalForms.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.EllipticCurve.VariableChange public import Mathlib.Algebra.CharP.Defs /-! - # Some normal forms of elliptic curves This file defines some normal forms of Weierstrass equations of elliptic curves. diff --git a/Mathlib/AlgebraicGeometry/Gluing.lean b/Mathlib/AlgebraicGeometry/Gluing.lean index 2c642da6af605a..358d6b4962eeb4 100644 --- a/Mathlib/AlgebraicGeometry/Gluing.lean +++ b/Mathlib/AlgebraicGeometry/Gluing.lean @@ -494,7 +494,6 @@ instance : (MorphismProperty.isomorphisms Scheme).IsLocalAtTarget zariskiPrecove simp, Cover.hom_ext (𝒰.pullback₁ f) _ _ fun i ↦ by simp, Cover.hom_ext 𝒰 _ _ fun i ↦ by simp⟩ /-! - ## Locally directed gluing We say that a diagram of open immersions is "locally directed" if for any `V, W ⊆ U` in the diagram, diff --git a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean index d32e5611a85a5b..316a0a8af51b3f 100644 --- a/Mathlib/AlgebraicGeometry/Modules/Tilde.lean +++ b/Mathlib/AlgebraicGeometry/Modules/Tilde.lean @@ -13,7 +13,6 @@ public import Mathlib.AlgebraicGeometry.Modules.Sheaf public import Mathlib.Data.Fintype.Order /-! - # Construction of M^~ Given any commutative ring `R` and `R`-module `M`, we construct the sheaf `M^~` of `𝒪_SpecR`-modules diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean index e2615dd23bd633..9c6ca6b27109b3 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Affine.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.IsIso public import Mathlib.AlgebraicGeometry.PullbackCarrier /-! - # Affine morphisms of schemes A morphism of schemes `f : X ⟶ Y` is affine if the preimage diff --git a/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean index 87d9a352ed1bba..d01a67f0cc06ed 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/ClosedImmersion.lean @@ -12,7 +12,6 @@ public import Mathlib.AlgebraicGeometry.ResidueField public import Mathlib.CategoryTheory.MorphismProperty.Comma /-! - # Closed immersions of schemes A morphism of schemes `f : X ⟶ Y` is a closed immersion if the underlying map of topological spaces diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean b/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean index 211e8505231f67..8bbf550642fddc 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Constructors.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.Basic public import Mathlib.RingTheory.RingHomProperties /-! - # Constructors for properties of morphisms between schemes This file provides some constructors to obtain morphism properties of schemes from other morphism diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Etale.lean b/Mathlib/AlgebraicGeometry/Morphisms/Etale.lean index 0b4f6f16974b8b..f37d48b9d2d1b8 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Etale.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Etale.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Smooth.StandardSmoothCotangent public import Mathlib.CategoryTheory.Limits.MorphismProperty /-! - # Étale morphisms A morphism of schemes `f : X ⟶ Y` is étale if for each affine `U ⊆ Y` diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Finite.lean b/Mathlib/AlgebraicGeometry/Morphisms/Finite.lean index e57a1ed93c8f18..a00474182be7f9 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Finite.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Finite.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Category.Ring.Epi public import Mathlib.RingTheory.Finiteness.Prod /-! - # Finite morphisms of schemes A morphism of schemes `f : X ⟶ Y` is finite if the preimage diff --git a/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean b/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean index ad50cc03d03afb..49fb8cac9ae67a 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/FinitePresentation.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.RingHom.FinitePresentation public import Mathlib.RingTheory.Spectrum.Prime.Chevalley /-! - # Morphisms of finite presentation A morphism of schemes `f : X ⟶ Y` is locally of finite presentation if for each affine `U ⊆ Y` and diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Immersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/Immersion.lean index 4bcef5bbce97af..59fd0ac4455b67 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Immersion.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Immersion.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.Separated public import Mathlib.AlgebraicGeometry.IdealSheaf.Functorial /-! - # Immersions of schemes A morphism of schemes `f : X ⟶ Y` is an immersion if the underlying map of topological spaces diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean b/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean index 925ae38a2fc25b..cbf9c2175e8e47 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Integral.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.UniversallyClosed public import Mathlib.RingTheory.RingHom.Integral /-! - # Integral morphisms of schemes A morphism of schemes `f : X ⟶ Y` is integral if the preimage diff --git a/Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean b/Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean index 45624f1d13c12d..a5794458bb5f24 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/IsIso.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicGeometry.Morphisms.OpenImmersion /-! - # Being an isomorphism is local at the target -/ diff --git a/Mathlib/AlgebraicGeometry/Morphisms/LocalFlatDescent.lean b/Mathlib/AlgebraicGeometry/Morphisms/LocalFlatDescent.lean index fb898294a78c3c..a89ab61eb9e25d 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/LocalFlatDescent.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/LocalFlatDescent.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.FlatDescent public import Mathlib.RingTheory.Etale.Descent /-! - # Local properties satisfying fpqc descent In this file we provide instances that show that the following local properties satisfy fpqc diff --git a/Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean index 8779719cf080a2..92065bfdd0eb21 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/OpenImmersion.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap /-! - # Open immersions A morphism is an open immersion if the underlying map of spaces is an open embedding diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean b/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean index 9344fdf1b8f994..f8956949125162 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Preimmersion.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap public import Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks /-! - # Preimmersions of schemes A morphism of schemes `f : X ⟶ Y` is a preimmersion if the underlying map of topological spaces diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Proper.lean b/Mathlib/AlgebraicGeometry/Morphisms/Proper.lean index 5baa0b7f663642..a1a6da6491c33d 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Proper.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Proper.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.Morphisms.Separated public import Mathlib.AlgebraicGeometry.Morphisms.Finite /-! - # Proper morphisms A morphism of schemes is proper if it is separated, universally closed and (locally) of finite type. diff --git a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean index 942a7ce2461eaa..31d0400264c382 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.LocalProperties.Basic public import Mathlib.RingTheory.RingHom.Locally /-! - # Properties of morphisms from properties of ring homs. We provide the basic framework for talking about properties of morphisms that come from properties diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Separated.lean b/Mathlib/AlgebraicGeometry/Morphisms/Separated.lean index 853b0ae33aa695..32672c1b66c94e 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Separated.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Separated.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Constructions.Over.Products public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer /-! - # Separated morphisms A morphism of schemes is separated if its diagonal morphism is a closed immersion. diff --git a/Mathlib/AlgebraicGeometry/Morphisms/Smooth.lean b/Mathlib/AlgebraicGeometry/Morphisms/Smooth.lean index c2148914765695..a1d8ca3b6e3f58 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/Smooth.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/Smooth.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.Smooth.Flat public import Mathlib.RingTheory.Smooth.Field /-! - # Smooth morphisms In this file we define smooth morphisms. The main definitions are: diff --git a/Mathlib/AlgebraicGeometry/Morphisms/WeaklyEtale.lean b/Mathlib/AlgebraicGeometry/Morphisms/WeaklyEtale.lean index 7059637e1bc0c2..8de358caddc419 100644 --- a/Mathlib/AlgebraicGeometry/Morphisms/WeaklyEtale.lean +++ b/Mathlib/AlgebraicGeometry/Morphisms/WeaklyEtale.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicGeometry.Morphisms.Etale /-! - # Weakly étale morphisms A morphism of schemes is weakly étale if it is flat and its diagonal is flat. As diff --git a/Mathlib/AlgebraicGeometry/PointsPi.lean b/Mathlib/AlgebraicGeometry/PointsPi.lean index fa54fcba2b584c..68ea34f1e69194 100644 --- a/Mathlib/AlgebraicGeometry/PointsPi.lean +++ b/Mathlib/AlgebraicGeometry/PointsPi.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicGeometry.Morphisms.Immersion /-! - # `Π Rᵢ`-Points of Schemes We show that the canonical map `X(Π Rᵢ) ⟶ Π X(Rᵢ)` (`AlgebraicGeometry.pointsPi`) diff --git a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean index b3d1dc0389c669..45641c2afc388a 100644 --- a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean +++ b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.AffineScheme public import Mathlib.AlgebraicGeometry.Gluing /-! - # Basic properties of the scheme `Proj A` The scheme `Proj 𝒜` for a graded ring `𝒜` is constructed in diff --git a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean index 90159f71e32c95..70e76bfe61ce0d 100644 --- a/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean +++ b/Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean @@ -44,7 +44,6 @@ Then we establish that `Proj 𝒜` is a `LocallyRingedSpace`: * [Robin Hartshorne, *Algebraic Geometry*][Har77] - -/ @[expose] public section diff --git a/Mathlib/AlgebraicGeometry/QuasiAffine.lean b/Mathlib/AlgebraicGeometry/QuasiAffine.lean index cd803f6d086d64..42ddf3ee74c65c 100644 --- a/Mathlib/AlgebraicGeometry/QuasiAffine.lean +++ b/Mathlib/AlgebraicGeometry/QuasiAffine.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicGeometry.Morphisms.Immersion /-! - # Quasi-affine schemes ## Main results diff --git a/Mathlib/AlgebraicGeometry/ResidueField.lean b/Mathlib/AlgebraicGeometry/ResidueField.lean index 20c400e69b5ca9..cd48cdef54e85e 100644 --- a/Mathlib/AlgebraicGeometry/ResidueField.lean +++ b/Mathlib/AlgebraicGeometry/ResidueField.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicGeometry.Stalk public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.ResidueField /-! - # Residue fields of points ## Main definitions @@ -25,7 +24,6 @@ The following are in the `AlgebraicGeometry.Scheme` namespace: - `AlgebraicGeometry.Scheme.SpecToEquivOfField`: morphisms `Spec K ⟶ X` for a field `K` correspond to pairs of `x : X` with embedding `κ(x) ⟶ K`. - -/ @[expose] public section diff --git a/Mathlib/AlgebraicGeometry/Sites/ElladicCohomology.lean b/Mathlib/AlgebraicGeometry/Sites/ElladicCohomology.lean index e41f93d88f48fe..955218b0189786 100644 --- a/Mathlib/AlgebraicGeometry/Sites/ElladicCohomology.lean +++ b/Mathlib/AlgebraicGeometry/Sites/ElladicCohomology.lean @@ -15,7 +15,6 @@ public import Mathlib.CategoryTheory.Sites.SheafCohomology.Basic public import Mathlib.NumberTheory.Padics.PadicIntegers /-! - # `ℓ`-adic cohomology of a scheme Let `X` be a scheme and `ℓ` be a prime number. In this file we define the sheaf diff --git a/Mathlib/AlgebraicGeometry/Sites/Etale.lean b/Mathlib/AlgebraicGeometry/Sites/Etale.lean index 6bcfcdeea6eb76..e9024aa06280ab 100644 --- a/Mathlib/AlgebraicGeometry/Sites/Etale.lean +++ b/Mathlib/AlgebraicGeometry/Sites/Etale.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Elements public import Mathlib.CategoryTheory.Sites.Point.Basic /-! - # The étale site In this file we define the big étale site, i.e. the étale topology as a Grothendieck topology diff --git a/Mathlib/AlgebraicGeometry/Sites/MorphismProperty.lean b/Mathlib/AlgebraicGeometry/Sites/MorphismProperty.lean index bc84f286e65bbc..a0f795650f658b 100644 --- a/Mathlib/AlgebraicGeometry/Sites/MorphismProperty.lean +++ b/Mathlib/AlgebraicGeometry/Sites/MorphismProperty.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.JointlySurjective public import Mathlib.CategoryTheory.Sites.MorphismProperty /-! - # Site defined by a morphism property Given a multiplicative morphism property `P` that is stable under base change, we define the diff --git a/Mathlib/AlgebraicGeometry/Sites/Proetale.lean b/Mathlib/AlgebraicGeometry/Sites/Proetale.lean index 8d8acd68952362..9634fa31dfa271 100644 --- a/Mathlib/AlgebraicGeometry/Sites/Proetale.lean +++ b/Mathlib/AlgebraicGeometry/Sites/Proetale.lean @@ -16,7 +16,6 @@ public import Mathlib.CategoryTheory.Sites.Subcanonical public import Mathlib.CategoryTheory.Sites.SubcanonicalOver /-! - # The pro-étale site In this file we define the big and small pro-étale site. The big pro-étale site is diff --git a/Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean b/Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean index 8b07f852c49337..9ef24ba9da5fbb 100644 --- a/Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean +++ b/Mathlib/AlgebraicGeometry/Sites/SmallAffineZariski.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicGeometry.RelativeGluing public import Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology /-! - # The small affine Zariski site `X.AffineZariskiSite` is the small affine Zariski site of `X`, whose elements are affine open diff --git a/Mathlib/AlgebraicGeometry/StructureSheaf.lean b/Mathlib/AlgebraicGeometry/StructureSheaf.lean index 065a8b4d8d8362..d1df0bb886e9bf 100644 --- a/Mathlib/AlgebraicGeometry/StructureSheaf.lean +++ b/Mathlib/AlgebraicGeometry/StructureSheaf.lean @@ -44,7 +44,6 @@ boundaries. * [Robin Hartshorne, *Algebraic Geometry*][Har77] - -/ diff --git a/Mathlib/AlgebraicGeometry/ZariskisMainTheorem.lean b/Mathlib/AlgebraicGeometry/ZariskisMainTheorem.lean index 0448888c13e6d0..81720da42be5ed 100644 --- a/Mathlib/AlgebraicGeometry/ZariskisMainTheorem.lean +++ b/Mathlib/AlgebraicGeometry/ZariskisMainTheorem.lean @@ -13,7 +13,6 @@ public import Mathlib.AlgebraicGeometry.Normalization public import Mathlib.RingTheory.Etale.QuasiFinite /-! - # Zariski's Main Theorem In this file we prove Grothendieck's reformulation of Zariski's main theorem, namely if diff --git a/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean b/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean index 7e6761fa8ed7d9..6abb3ed7b10914 100644 --- a/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean +++ b/Mathlib/AlgebraicTopology/AlternatingFaceMapComplex.lean @@ -13,7 +13,6 @@ public import Mathlib.CategoryTheory.Limits.FunctorCategory.EpiMono public import Mathlib.CategoryTheory.Preadditive.Opposite /-! - # The alternating face map complex of a simplicial object in a preadditive category We construct the alternating face map complex, as a diff --git a/Mathlib/AlgebraicTopology/CechNerve.lean b/Mathlib/AlgebraicTopology/CechNerve.lean index 9b9f4f0b1aff3e..fd67341de55458 100644 --- a/Mathlib/AlgebraicTopology/CechNerve.lean +++ b/Mathlib/AlgebraicTopology/CechNerve.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts public import Mathlib.Tactic.ApplyFun /-! - # The Čech Nerve This file provides a definition of the Čech nerve associated to an arrow, provided diff --git a/Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean b/Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean index 68becbab092b2e..fcc6485041f686 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.DoldKan.PInfty /-! - # Decomposition of the Q endomorphisms In this file, we obtain a lemma `decomposition_Q` which expresses diff --git a/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean b/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean index 5601ba9c94caba..18faa1217429b7 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean @@ -10,7 +10,6 @@ public import Mathlib.Tactic.FinCases public import Mathlib.Tactic.Linarith /-! - # Behaviour of `P_infty` with respect to degeneracies For any `X : SimplicialObject C` where `C` is an abelian category, diff --git a/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean b/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean index 97716a6dda8fd9..defdab1a3cc2bd 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Equivalence.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicTopology.DoldKan.EquivalencePseudoabelian public import Mathlib.AlgebraicTopology.DoldKan.Normalized /-! - # The Dold-Kan correspondence The Dold-Kan correspondence states that for any abelian category `A`, there is diff --git a/Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean b/Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean index 1e7257e5e2d02f..4ae01eb616586f 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/EquivalencePseudoabelian.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Idempotents.SimplicialObject public import Mathlib.Tactic.SuppressCompilation /-! - # The Dold-Kan correspondence for pseudoabelian categories In this file, for any idempotent complete additive category `C`, diff --git a/Mathlib/AlgebraicTopology/DoldKan/Faces.lean b/Mathlib/AlgebraicTopology/DoldKan/Faces.lean index 6cab6acea43b2e..ea5a05fac7e4ce 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Faces.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Faces.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicTopology.DoldKan.Homotopies public import Mathlib.Tactic.Ring /-! - # Study of face maps for the Dold-Kan correspondence In this file, we obtain the technical lemmas that are used in the file diff --git a/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean b/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean index f6a735cfaa3bd1..c584890e429df2 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean @@ -9,10 +9,8 @@ public import Mathlib.AlgebraicTopology.SimplicialObject.Split public import Mathlib.AlgebraicTopology.DoldKan.PInfty /-! - # Construction of the inverse functor of the Dold-Kan equivalence - In this file, we construct the functor `Γ₀ : ChainComplex C ℕ ⥤ SimplicialObject C` which shall be the inverse functor of the Dold-Kan equivalence in the case of abelian categories, and more generally pseudoabelian categories. diff --git a/Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean b/Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean index cbfd75b4f86e11..bf0e94507397a5 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/FunctorN.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.DoldKan.PInfty /-! - # Construction of functors N for the Dold-Kan correspondence In this file, we construct functors `N₁ : SimplicialObject C ⥤ Karoubi (ChainComplex C ℕ)` diff --git a/Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean b/Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean index 544576f07024a4..38621665b155ed 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Homology.Homotopy public import Mathlib.AlgebraicTopology.DoldKan.Notations /-! - # Construction of homotopies for the Dold-Kan correspondence (The general strategy of proof of the Dold-Kan correspondence is explained diff --git a/Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean b/Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean index 0110b8af599dbe..a3d11ed9d1e5d7 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/HomotopyEquivalence.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.DoldKan.Normalized /-! - # The normalized Moore complex and the alternating face map complex are homotopy equivalent In this file, when the category `A` is abelian, we obtain the homotopy equivalence diff --git a/Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean b/Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean index b73fc26d9b4b64..4363d7d29896d9 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/NReflectsIso.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Idempotents.HomologicalComplex public import Mathlib.CategoryTheory.Idempotents.KaroubiKaroubi /-! - # N₁ and N₂ reflect isomorphisms In this file, it is shown that the functors diff --git a/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean b/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean index 43c7aa64499ae2..1e1f839392d80d 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Normalized.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.DoldKan.FunctorN /-! - # Comparison with the normalized Moore complex functor In this file, we show that when the category `A` is abelian, diff --git a/Mathlib/AlgebraicTopology/DoldKan/Notations.lean b/Mathlib/AlgebraicTopology/DoldKan/Notations.lean index 538cadceeebf07..0573ad9f22f941 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Notations.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Notations.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.AlternatingFaceMapComplex /-! - # Notations for the Dold-Kan equivalence This file defines the notation `K[X] : ChainComplex C ℕ` for the alternating face diff --git a/Mathlib/AlgebraicTopology/DoldKan/PInfty.lean b/Mathlib/AlgebraicTopology/DoldKan/PInfty.lean index c5263655e2e9b3..d153e732398e84 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/PInfty.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/PInfty.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Idempotents.FunctorCategories public import Mathlib.CategoryTheory.Idempotents.FunctorExtension /-! - # Construction of the projection `PInfty` for the Dold-Kan correspondence In this file, we construct the projection `PInfty : K[X] ⟶ K[X]` by passing diff --git a/Mathlib/AlgebraicTopology/DoldKan/Projections.lean b/Mathlib/AlgebraicTopology/DoldKan/Projections.lean index e960ff9b0464b6..0acbc6dac26509 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/Projections.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/Projections.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicTopology.DoldKan.Faces public import Mathlib.CategoryTheory.Idempotents.Basic /-! - # Construction of projections for the Dold-Kan correspondence In this file, we construct endomorphisms `P q : K[X] ⟶ K[X]` for all diff --git a/Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean b/Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean index f91f096b7b636f..dd0a3669d01dba 100644 --- a/Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean +++ b/Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean @@ -10,7 +10,6 @@ public import Mathlib.AlgebraicTopology.DoldKan.HomotopyEquivalence public import Mathlib.AlgebraicTopology.SimplicialObject.Split /-! - # Split simplicial objects in preadditive categories In this file we define a functor `nondegComplex : SimplicialObject.Split C ⥤ ChainComplex C ℕ` diff --git a/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean b/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean index a383cecb67724a..69e8160b7bea24 100644 --- a/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean +++ b/Mathlib/AlgebraicTopology/ExtraDegeneracy.lean @@ -13,7 +13,6 @@ public import Mathlib.AlgebraicTopology.SimplicialObject.Homotopy public import Mathlib.AlgebraicTopology.SimplicialSet.StdSimplex /-! - # Augmented simplicial objects with an extra degeneracy In simplicial homotopy theory, in order to prove that the connected components diff --git a/Mathlib/AlgebraicTopology/SimplicialNerve.lean b/Mathlib/AlgebraicTopology/SimplicialNerve.lean index e09a994f76df67..ea9bafbcb513b8 100644 --- a/Mathlib/AlgebraicTopology/SimplicialNerve.lean +++ b/Mathlib/AlgebraicTopology/SimplicialNerve.lean @@ -8,7 +8,6 @@ module public import Mathlib.AlgebraicTopology.SimplicialCategory.Basic public import Mathlib.AlgebraicTopology.SimplicialSet.Nerve /-! - # The simplicial nerve of a simplicial category This file defines the simplicial nerve (sometimes called homotopy coherent nerve) of a simplicial diff --git a/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean b/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean index ec40d99b4bf5e3..546c78e637cf55 100644 --- a/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean +++ b/Mathlib/AlgebraicTopology/SimplicialObject/Split.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products public import Mathlib.Data.Fintype.Sigma /-! - # Split simplicial objects In this file, we introduce the notion of split simplicial object. diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Rank.lean b/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Rank.lean index 28299b8bd37ccd..4f1bb5078122a2 100644 --- a/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Rank.lean +++ b/Mathlib/AlgebraicTopology/SimplicialSet/AnodyneExtensions/Rank.lean @@ -23,7 +23,6 @@ then `P.RankFunction ℕ` is non empty (TODO @joelriou). (We also introduce similar definitions for the structure `PairingCore`.) - ## References * [Sean Moss, *Another approach to the Kan-Quillen model structure*][moss-2020] diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean b/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean index 9f02d77707a461..0d422821fea514 100644 --- a/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean +++ b/Mathlib/AlgebraicTopology/SimplicialSet/Boundary.lean @@ -20,7 +20,6 @@ As an example, we should have a function that constructs from a non-surjective order-preserving function `Fin n → Fin n` a morphism `Δ[n] ⟶ ∂Δ[n]`. - -/ @[expose] public section diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean b/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean index 808bbc3b83a4c1..69b3d16ea528db 100644 --- a/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean +++ b/Mathlib/AlgebraicTopology/SimplicialSet/HomotopyCat.lean @@ -15,7 +15,6 @@ public import Mathlib.AlgebraicTopology.SimplicialSet.Monoidal public import Mathlib.CategoryTheory.Category.Cat.Terminal /-! - # The homotopy category of a simplicial set The homotopy category of a simplicial set is defined as a quotient of the free category on its diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/Nerve.lean b/Mathlib/AlgebraicTopology/SimplicialSet/Nerve.lean index b8867db3e13b61..cac232b19540cd 100644 --- a/Mathlib/AlgebraicTopology/SimplicialSet/Nerve.lean +++ b/Mathlib/AlgebraicTopology/SimplicialSet/Nerve.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicTopology.SimplicialSet.CompStruct public import Mathlib.CategoryTheory.ComposableArrows.Basic /-! - # The nerve of a category This file provides the definition of the nerve of a category `C`, diff --git a/Mathlib/AlgebraicTopology/SimplicialSet/Op.lean b/Mathlib/AlgebraicTopology/SimplicialSet/Op.lean index c470f8e3566c1f..319f17ad406355 100644 --- a/Mathlib/AlgebraicTopology/SimplicialSet/Op.lean +++ b/Mathlib/AlgebraicTopology/SimplicialSet/Op.lean @@ -16,7 +16,6 @@ of the category of simplicial sets that is induced by the covariant involution `SimplexCategory.op : SimplexCategory ⥤ SimplexCategory`. We use an abbreviation `X.op` for `opFunctor.obj X`. - ## TODO * Show that this involution sends `Δ[n]` to itself, and that via diff --git a/Mathlib/Analysis/Analytic/Inverse.lean b/Mathlib/Analysis/Analytic/Inverse.lean index 5a1ae3790ef0d7..d918b9b0a3a37b 100644 --- a/Mathlib/Analysis/Analytic/Inverse.lean +++ b/Mathlib/Analysis/Analytic/Inverse.lean @@ -10,7 +10,6 @@ public import Mathlib.Analysis.Analytic.Linear public import Mathlib.Tactic.Positivity /-! - # Inverse of analytic functions We construct the left and right inverse of a formal multilinear series with invertible linear term, diff --git a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Continuity.lean b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Continuity.lean index e423b7abf9d708..86790fb684421d 100644 --- a/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Continuity.lean +++ b/Mathlib/Analysis/CStarAlgebra/ContinuousFunctionalCalculus/Continuity.lean @@ -39,7 +39,6 @@ results in the variable `a` come in two flavors: those for `RCLike 𝕜` and tho ## Main results - + `tendsto_cfc_fun`: If `F : X → R → R` tends to `f : R → R` uniformly on the spectrum of `a`, and all these functions are continuous on the spectrum, then `fun x ↦ cfc (F x) a` tends to `cfc f a`. diff --git a/Mathlib/Analysis/CStarAlgebra/Projection.lean b/Mathlib/Analysis/CStarAlgebra/Projection.lean index e05ce8f90d7be2..d97561ec7a1da6 100644 --- a/Mathlib/Analysis/CStarAlgebra/Projection.lean +++ b/Mathlib/Analysis/CStarAlgebra/Projection.lean @@ -11,7 +11,6 @@ public import Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpo import Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Projection /-! - # Projections in C⋆-algebras Here we collect results about projections specific to C⋆-algebras. diff --git a/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean index 084664a10b855d..2ae2b510152112 100644 --- a/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean +++ b/Mathlib/Analysis/Calculus/ContDiff/FTaylorSeries.lean @@ -42,7 +42,6 @@ Let `f : E → F` be a map between normed vector spaces over a nontrivially norm It is a continuous multilinear map from `E^n` to `F`, defined as a derivative of `iteratedFDeriv 𝕜 (n-1) f` if one exists, and `0` otherwise. - ### Side of the composition, and universe issues With a naïve direct definition, the `n`-th derivative of a function belongs to the space diff --git a/Mathlib/Analysis/Calculus/Deriv/Basic.lean b/Mathlib/Analysis/Calculus/Deriv/Basic.lean index 89412bdad756e9..352994b212c90a 100644 --- a/Mathlib/Analysis/Calculus/Deriv/Basic.lean +++ b/Mathlib/Analysis/Calculus/Deriv/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Analysis.Calculus.TangentCone.Real public import Mathlib.Analysis.Normed.Operator.Bilinear /-! - # One-dimensional derivatives This file defines the derivative of a function `f : 𝕜 → F` where `𝕜` is a diff --git a/Mathlib/Analysis/Calculus/Deriv/MeanValue.lean b/Mathlib/Analysis/Calculus/Deriv/MeanValue.lean index bb2d7794ab3b86..6b1380ae4e0053 100644 --- a/Mathlib/Analysis/Calculus/Deriv/MeanValue.lean +++ b/Mathlib/Analysis/Calculus/Deriv/MeanValue.lean @@ -40,7 +40,6 @@ is not differentiable on the right at that point, and similarly differentiabilit ## Main results - * `exists_ratio_hasDerivAt_eq_ratio_slope` and `exists_ratio_deriv_eq_ratio_slope` : Cauchy's Mean Value Theorem. diff --git a/Mathlib/Analysis/Calculus/DerivativeTest.lean b/Mathlib/Analysis/Calculus/DerivativeTest.lean index 376b6428c73e56..6e4c2c311cae87 100644 --- a/Mathlib/Analysis/Calculus/DerivativeTest.lean +++ b/Mathlib/Analysis/Calculus/DerivativeTest.lean @@ -41,7 +41,6 @@ Source: [Wikipedia](https://en.wikipedia.org/wiki/Derivative_test#Proof_of_the_s * `isLocalMin_of_deriv_deriv_pos`: The second-derivative test, minimum version. - ## Tags derivative test, first-derivative test, second-derivative test, calculus diff --git a/Mathlib/Analysis/Calculus/ParametricIntegral.lean b/Mathlib/Analysis/Calculus/ParametricIntegral.lean index a6a6365b699152..a2f609130d01a2 100644 --- a/Mathlib/Analysis/Calculus/ParametricIntegral.lean +++ b/Mathlib/Analysis/Calculus/ParametricIntegral.lean @@ -22,7 +22,6 @@ to guarantee that `f` is continuous using the dominated convergence theorem. In we want to express the derivative of `f` as the integral of the derivative of `F` with respect to `x`. - ## Main results As explained above, all results express the derivative of a parametric integral as the integral of diff --git a/Mathlib/Analysis/Complex/Basic.lean b/Mathlib/Analysis/Complex/Basic.lean index ded2fe18b25a89..3b1c114601eac5 100644 --- a/Mathlib/Analysis/Complex/Basic.lean +++ b/Mathlib/Analysis/Complex/Basic.lean @@ -15,7 +15,6 @@ public import Mathlib.Topology.Algebra.Module.ContinuousLinearMap.RestrictScalar public import Mathlib.Topology.Instances.RealVectorSpace /-! - # Normed space structure on `ℂ`. This file gathers basic facts of analytic nature on the complex numbers. diff --git a/Mathlib/Analysis/Convex/MetricSpace.lean b/Mathlib/Analysis/Convex/MetricSpace.lean index 848ed2f4e6d82d..48ca5517306d08 100644 --- a/Mathlib/Analysis/Convex/MetricSpace.lean +++ b/Mathlib/Analysis/Convex/MetricSpace.lean @@ -12,7 +12,6 @@ public import Mathlib.Geometry.Convex.ConvexSpace.AffineSpace public import Mathlib.Geometry.Convex.ConvexSpace.Module /-! - # Convex spaces with compatible metric structure A convex space has a compatible metric structure if `dist(∑ tᵢ xᵢ, ∑ tᵢ yᵢ) ≤ ∑ tᵢ dist(xᵢ, yᵢ)`. diff --git a/Mathlib/Analysis/Distribution/Sobolev.lean b/Mathlib/Analysis/Distribution/Sobolev.lean index 3a49f003320e3a..db386b1291562c 100644 --- a/Mathlib/Analysis/Distribution/Sobolev.lean +++ b/Mathlib/Analysis/Distribution/Sobolev.lean @@ -40,7 +40,6 @@ the definition of the Sobolev spaces. * `TemperedDistribution.MemSobolev.laplacian`: If `u` is a Sobolev function of order `s`, then `Δ u` is a Sobolev function of order `s - 2`. - ## References * [M. Taylor, *Partial Differential Equations 1*][taylorPDE1] * [W. McLean, *Strongly Elliptic Systems and Boundary Integral Equations*][mclean2000] diff --git a/Mathlib/Analysis/Fourier/AddCircle.lean b/Mathlib/Analysis/Fourier/AddCircle.lean index 36d822059852c1..bced999c11e53e 100644 --- a/Mathlib/Analysis/Fourier/AddCircle.lean +++ b/Mathlib/Analysis/Fourier/AddCircle.lean @@ -16,7 +16,6 @@ public import Mathlib.Topology.ContinuousMap.StoneWeierstrass public import Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts /-! - # Fourier analysis on the additive circle This file contains basic results on Fourier series for functions on the additive circle diff --git a/Mathlib/Analysis/Fourier/LpSpace.lean b/Mathlib/Analysis/Fourier/LpSpace.lean index 5475d647eb65ba..5563dfc7445fb6 100644 --- a/Mathlib/Analysis/Fourier/LpSpace.lean +++ b/Mathlib/Analysis/Fourier/LpSpace.lean @@ -9,7 +9,6 @@ public import Mathlib.Analysis.Distribution.TemperedDistribution public import Mathlib.Analysis.Normed.Operator.Extend /-! - # The Fourier transform on $L^p$ In this file we define the Fourier transform on $L^2$ as a linear isometry equivalence. diff --git a/Mathlib/Analysis/InnerProductSpace/Dual.lean b/Mathlib/Analysis/InnerProductSpace/Dual.lean index 03916c4cfa434e..b73880a2498695 100644 --- a/Mathlib/Analysis/InnerProductSpace/Dual.lean +++ b/Mathlib/Analysis/InnerProductSpace/Dual.lean @@ -26,7 +26,6 @@ For a bounded sesquilinear form `B : E →L⋆[𝕜] E →L[𝕜] 𝕜`, we define a map `InnerProductSpace.continuousLinearMapOfBilin B : E →L[𝕜] E`, given by substituting `E →L[𝕜] 𝕜` with `E` using `toDual`. - ## References * [M. Einsiedler and T. Ward, *Functional Analysis, Spectral Theory, and Applications*] diff --git a/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean b/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean index da112f39df47a6..ab1ed5113dbb0c 100644 --- a/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean +++ b/Mathlib/Analysis/InnerProductSpace/LinearPMap.lean @@ -12,7 +12,6 @@ public import Mathlib.Topology.Algebra.Module.Equiv public import Mathlib.Topology.Algebra.Module.LinearPMap /-! - # Partially defined linear operators on Hilbert spaces We will develop the basics of the theory of unbounded operators on Hilbert spaces. diff --git a/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean b/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean index e2360e5c1d9801..1a113fa78f8b97 100644 --- a/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean +++ b/Mathlib/Analysis/InnerProductSpace/TensorProduct.lean @@ -12,7 +12,6 @@ import Mathlib.Analysis.InnerProductSpace.GramMatrix import Mathlib.Analysis.InnerProductSpace.Positive /-! - # Inner product space structure on tensor product spaces This file provides the inner product space structure on tensor product spaces. diff --git a/Mathlib/Analysis/MeanInequalities.lean b/Mathlib/Analysis/MeanInequalities.lean index b6aba130f375d6..c3e5cd9ed56175 100644 --- a/Mathlib/Analysis/MeanInequalities.lean +++ b/Mathlib/Analysis/MeanInequalities.lean @@ -41,7 +41,6 @@ a version for real-valued non-negative functions is in the `Real` namespace, and - `geom_mean_le_arith_mean3_weighted` : weighted version for three numbers; - `geom_mean_le_arith_mean4_weighted` : weighted version for four numbers. - ### HM-GM inequality: The inequality says that the harmonic mean of a tuple of positive numbers is less than or equal diff --git a/Mathlib/Analysis/Normed/Group/Quotient.lean b/Mathlib/Analysis/Normed/Group/Quotient.lean index a2be9239e6f3e8..a7e46460015001 100644 --- a/Mathlib/Analysis/Normed/Group/Quotient.lean +++ b/Mathlib/Analysis/Normed/Group/Quotient.lean @@ -37,7 +37,6 @@ this instance in `Submodule.Quotient.completeSpace` so that it applies to these ## Main definitions - We use `M` and `N` to denote seminormed groups and `S : AddSubgroup M`. All the following definitions are in the `AddSubgroup` namespace. Hence we can access `AddSubgroup.normedMk S` as `S.normedMk`. @@ -65,7 +64,6 @@ All the following definitions are in the `AddSubgroup` namespace. Hence we can a * `IsQuotient.norm_lift`: Provided `f : normed_hom M N` satisfies `IsQuotient f`, for every `n : N` and positive `ε`, there exists `m` such that `f m = n ∧ ‖m‖ < ‖n‖ + ε`. - ## Implementation details For any `SeminormedAddCommGroup M` and any `S : AddSubgroup M` we define a norm on `M ⧸ S` by diff --git a/Mathlib/Analysis/Normed/Lp/ProdLp.lean b/Mathlib/Analysis/Normed/Lp/ProdLp.lean index 739a0034d95587..6a39fec065092e 100644 --- a/Mathlib/Analysis/Normed/Lp/ProdLp.lean +++ b/Mathlib/Analysis/Normed/Lp/ProdLp.lean @@ -35,7 +35,6 @@ statements for the coordinate functions, for instance. If you wish to endow a type synonym of `α × β` with the `L^p` distance, you can use `pseudoMetricSpaceToProd` and the declarations below that one. - ## Implementation notes This file is a straight-forward adaptation of `Mathlib/Analysis/Normed/Lp/PiLp.lean`. diff --git a/Mathlib/Analysis/Normed/Lp/SmoothApprox.lean b/Mathlib/Analysis/Normed/Lp/SmoothApprox.lean index be25da2e7b60e1..fb2c50d08678a1 100644 --- a/Mathlib/Analysis/Normed/Lp/SmoothApprox.lean +++ b/Mathlib/Analysis/Normed/Lp/SmoothApprox.lean @@ -13,7 +13,6 @@ import Mathlib.Geometry.Manifold.SmoothApprox import Mathlib.MeasureTheory.Function.ContinuousMapDense /-! - # Density of smooth compactly supported functions in `Lp` In this file, we prove that `Lp` functions can be approximated by smooth compactly supported diff --git a/Mathlib/Analysis/Normed/Operator/Conformal.lean b/Mathlib/Analysis/Normed/Operator/Conformal.lean index 68bbcd478affcf..df782c25994db3 100644 --- a/Mathlib/Analysis/Normed/Operator/Conformal.lean +++ b/Mathlib/Analysis/Normed/Operator/Conformal.lean @@ -28,7 +28,6 @@ See `Analysis.InnerProductSpace.ConformalLinearMap` for * `isConformalMap_iff`: a map between inner product spaces is conformal iff it preserves inner products up to a fixed scalar factor. - ## Tags conformal diff --git a/Mathlib/Analysis/Normed/Operator/Extend.lean b/Mathlib/Analysis/Normed/Operator/Extend.lean index 69a0fdf95e9e18..76b483403ea81d 100644 --- a/Mathlib/Analysis/Normed/Operator/Extend.lean +++ b/Mathlib/Analysis/Normed/Operator/Extend.lean @@ -9,7 +9,6 @@ public import Mathlib.Analysis.Normed.Operator.Basic public import Mathlib.LinearAlgebra.Isomorphisms /-! - # Extension of continuous linear maps on Banach spaces In this file we provide two different ways to extend a continuous linear map defined on a dense diff --git a/Mathlib/Analysis/Normed/Unbundled/SeminormFromBounded.lean b/Mathlib/Analysis/Normed/Unbundled/SeminormFromBounded.lean index 2c06c768cb8bc9..c3bdbb67ff044d 100644 --- a/Mathlib/Analysis/Normed/Unbundled/SeminormFromBounded.lean +++ b/Mathlib/Analysis/Normed/Unbundled/SeminormFromBounded.lean @@ -17,7 +17,6 @@ the inequality `f (x * y) ≤ c * f x * f y)` is satisfied, we create a ring sem In the file comments, we will use the expression `f is multiplicatively bounded` to indicate that this condition holds. - ## Main Definitions * `seminormFromBounded'` : the real-valued function sending `x ∈ R` to the supremum of @@ -27,7 +26,6 @@ this condition holds. nonnegative, multiplicatively bounded and subadditive, that it preserves `0` and negation, and that `f` has trivial kernel. - ## Main Results * `seminormFromBounded_isNonarchimedean` : if `f : R → ℝ` is a nonnegative, multiplicatively diff --git a/Mathlib/Analysis/Normed/Unbundled/SeminormFromConst.lean b/Mathlib/Analysis/Normed/Unbundled/SeminormFromConst.lean index 86c4d3235960ae..e66ae3e9a1e311 100644 --- a/Mathlib/Analysis/Normed/Unbundled/SeminormFromConst.lean +++ b/Mathlib/Analysis/Normed/Unbundled/SeminormFromConst.lean @@ -10,7 +10,6 @@ public import Mathlib.Analysis.Normed.Unbundled.RingSeminorm /-! # SeminormFromConst - In this file, we prove [BGR, Proposition 1.3.2/2][bosch-guntzer-remmert] : starting from a power-multiplicative seminorm on a commutative ring `R` and a nonzero `c : R`, we create a new power-multiplicative seminorm for which `c` is multiplicative. @@ -21,7 +20,6 @@ power-multiplicative seminorm for which `c` is multiplicative. `(f (x * c^n))/((f c)^n)`. * `seminormFromConst` : the function `seminormFromConst'` as a `RingSeminorm` on `R`. - ## Main Results * `seminormFromConst_isNonarchimedean` : the function `seminormFromConst' c f` is nonarchimedean when f is nonarchimedean. diff --git a/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean b/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean index c8788f4428e684..07eddfaa3b131c 100644 --- a/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean +++ b/Mathlib/Analysis/SpecialFunctions/Elliptic/Weierstrass.lean @@ -16,7 +16,6 @@ public import Mathlib.Topology.Algebra.InfiniteSum.UniformOn public import Mathlib.Topology.MetricSpace.ProperSpace.Lemmas /-! - # Weierstrass `℘` functions ## Main definitions and results diff --git a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean index 8e4f0f73a84f83..4bacebd581c583 100644 --- a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean +++ b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory public import Mathlib.CategoryTheory.Limits.Shapes.Countable public import Mathlib.Logic.Equiv.List /-! - # Grothendieck Axioms This file defines some of the Grothendieck Axioms for abelian categories, and proves diff --git a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/FunctorCategory.lean b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/FunctorCategory.lean index f8750cb0703e08..5e38894cf90dff 100644 --- a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/FunctorCategory.lean +++ b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/FunctorCategory.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Basic /-! - # AB axioms in functor categories This file proves that, when the relevant limits and colimits exist, exactness of limits and diff --git a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Sheaf.lean b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Sheaf.lean index 2601697678b5ed..37c0b5f346a32f 100644 --- a/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Sheaf.lean +++ b/Mathlib/CategoryTheory/Abelian/GrothendieckAxioms/Sheaf.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Sites.Abelian public import Mathlib.CategoryTheory.Sites.Equivalence /-! - # AB axioms in sheaf categories If `J` is a Grothendieck topology on a small category `C : Type v`, diff --git a/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Basic.lean b/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Basic.lean index 8fda662a786c80..69296fbe57d906 100644 --- a/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Basic.lean +++ b/Mathlib/CategoryTheory/Abelian/GrothendieckCategory/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Adjunction.AdjointFunctorTheorems public import Mathlib.CategoryTheory.Limits.HasLimits /-! - # Grothendieck categories This file defines Grothendieck categories and proves basic facts about them. diff --git a/Mathlib/CategoryTheory/Abelian/Preradical/Radical.lean b/Mathlib/CategoryTheory/Abelian/Preradical/Radical.lean index 5ad3bb3804ed11..3214d7a44cf6dc 100644 --- a/Mathlib/CategoryTheory/Abelian/Preradical/Radical.lean +++ b/Mathlib/CategoryTheory/Abelian/Preradical/Radical.lean @@ -21,7 +21,6 @@ We encode this as the property that the natural transformation `toColon Φ Φ : is an isomorphism, and we prove a basic characterization of radicals in terms of the vanishing of `Φ.r` on `Φ.quotient`. - ## Main definitions * `Preradical.IsRadical` : diff --git a/Mathlib/CategoryTheory/Action/Continuous.lean b/Mathlib/CategoryTheory/Action/Continuous.lean index b5da64b91f2320..52102c04a32167 100644 --- a/Mathlib/CategoryTheory/Action/Continuous.lean +++ b/Mathlib/CategoryTheory/Action/Continuous.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Category.TopCat.Basic public import Mathlib.Topology.Algebra.ContinuousMonoidHom /-! - # Topological subcategories of `Action V G` For a concrete category `V`, where the forgetful functor factors via `TopCat`, diff --git a/Mathlib/CategoryTheory/Adhesive/Basic.lean b/Mathlib/CategoryTheory/Adhesive/Basic.lean index b10b4cf4c2cf85..595d77b0fae434 100644 --- a/Mathlib/CategoryTheory/Adhesive/Basic.lean +++ b/Mathlib/CategoryTheory/Adhesive/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.KernelPair public import Mathlib.CategoryTheory.Limits.Constructions.EpiMono /-! - # Adhesive categories ## Main definitions diff --git a/Mathlib/CategoryTheory/Adhesive/Subobject.lean b/Mathlib/CategoryTheory/Adhesive/Subobject.lean index b1b154346e4a16..723cf0bf83ada5 100644 --- a/Mathlib/CategoryTheory/Adhesive/Subobject.lean +++ b/Mathlib/CategoryTheory/Adhesive/Subobject.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Adhesive.Basic public import Mathlib.CategoryTheory.Subobject.Basic /-! - # Subobjects in adhesive categories ## Main Results diff --git a/Mathlib/CategoryTheory/Adjunction/Evaluation.lean b/Mathlib/CategoryTheory/Adjunction/Evaluation.lean index 1681bf65fc437d..a4d25defdebaee 100644 --- a/Mathlib/CategoryTheory/Adjunction/Evaluation.lean +++ b/Mathlib/CategoryTheory/Adjunction/Evaluation.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Limits.Shapes.Products /-! - # Adjunctions involving evaluation We show that evaluation of functors has adjoints, given the existence of (co)products. diff --git a/Mathlib/CategoryTheory/Adjunction/Restrict.lean b/Mathlib/CategoryTheory/Adjunction/Restrict.lean index 416187d5e84c72..a7a3fd65e6513b 100644 --- a/Mathlib/CategoryTheory/Adjunction/Restrict.lean +++ b/Mathlib/CategoryTheory/Adjunction/Restrict.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Adjunction.Basic public import Mathlib.CategoryTheory.HomCongr /-! - # Restricting adjunctions `Adjunction.restrictFullyFaithful` shows that an adjunction can be restricted along fully faithful diff --git a/Mathlib/CategoryTheory/Adjunction/Triple.lean b/Mathlib/CategoryTheory/Adjunction/Triple.lean index 952f2a5bf6b9b6..ffbae5242eeb7b 100644 --- a/Mathlib/CategoryTheory/Adjunction/Triple.lean +++ b/Mathlib/CategoryTheory/Adjunction/Triple.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Adjunction.Opposites public import Mathlib.CategoryTheory.Adjunction.Unique public import Mathlib.CategoryTheory.Monad.Adjunction /-! - # Adjoint triples This file concerns adjoint triples `F ⊣ G ⊣ H` of functors `F H : C ⥤ D`, `G : D ⥤ C`. We first diff --git a/Mathlib/CategoryTheory/Adjunction/Unique.lean b/Mathlib/CategoryTheory/Adjunction/Unique.lean index 0cd61c22d58863..0c4e1f4140f4fa 100644 --- a/Mathlib/CategoryTheory/Adjunction/Unique.lean +++ b/Mathlib/CategoryTheory/Adjunction/Unique.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Adjunction.Mates /-! - # Uniqueness of adjoints This file shows that adjoints are unique up to natural isomorphism. diff --git a/Mathlib/CategoryTheory/Adjunction/Whiskering.lean b/Mathlib/CategoryTheory/Adjunction/Whiskering.lean index eb7c9f90b6b336..398edd02d363c0 100644 --- a/Mathlib/CategoryTheory/Adjunction/Whiskering.lean +++ b/Mathlib/CategoryTheory/Adjunction/Whiskering.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Whiskering public import Mathlib.CategoryTheory.Adjunction.Basic /-! - Given categories `C D E`, functors `F : D ⥤ E` and `G : E ⥤ D` with an adjunction `F ⊣ G`, we provide the induced adjunction between the functor categories `C ⥤ D` and `C ⥤ E`, and the functor categories `E ⥤ C` and `D ⥤ C`. diff --git a/Mathlib/CategoryTheory/Bicategory/Adjunction/Mate.lean b/Mathlib/CategoryTheory/Bicategory/Adjunction/Mate.lean index 49d6e2eae87ab8..5ab62a7e970f56 100644 --- a/Mathlib/CategoryTheory/Bicategory/Adjunction/Mate.lean +++ b/Mathlib/CategoryTheory/Bicategory/Adjunction/Mate.lean @@ -27,7 +27,6 @@ For the bicategory `Cat`, the definitions in this file are provided in `Mathlib/CategoryTheory/Adjunction/Mates.lean`, where you can find more detailed documentation about mates. - ## Implementation The correspondence between mates is obtained by combining diff --git a/Mathlib/CategoryTheory/Bicategory/Functor/Prelax.lean b/Mathlib/CategoryTheory/Bicategory/Functor/Prelax.lean index 653a26fe111dae..e2c0f6378e418b 100644 --- a/Mathlib/CategoryTheory/Bicategory/Functor/Prelax.lean +++ b/Mathlib/CategoryTheory/Bicategory/Functor/Prelax.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Bicategory.Basic public import Mathlib.CategoryTheory.EqToHom /-! - # Prelax functors This file defines lax prefunctors and prelax functors between bicategories. The point of these diff --git a/Mathlib/CategoryTheory/Bicategory/InducedBicategory.lean b/Mathlib/CategoryTheory/Bicategory/InducedBicategory.lean index 687fd8e8373703..071b3bbe340faf 100644 --- a/Mathlib/CategoryTheory/Bicategory/InducedBicategory.lean +++ b/Mathlib/CategoryTheory/Bicategory/InducedBicategory.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Bicategory.Functor.StrictPseudofunctor /-! - # Induced bicategories In this file we develop API for constructing a full sub-bicategory of a bicategory `C`, given a diff --git a/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.lean b/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.lean index 9a14a4bb255ac5..8a96ccc337a085 100644 --- a/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.lean +++ b/Mathlib/CategoryTheory/Bicategory/NaturalTransformation/Pseudo.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor public import Mathlib.CategoryTheory.Bicategory.NaturalTransformation.Oplax /-! - # Strong transformations of pseudofunctors There are three types of transformations between pseudofunctors, depending on the direction diff --git a/Mathlib/CategoryTheory/Category/GaloisConnection.lean b/Mathlib/CategoryTheory/Category/GaloisConnection.lean index e44c989e87fe9e..ec1e1ee24b5f7f 100644 --- a/Mathlib/CategoryTheory/Category/GaloisConnection.lean +++ b/Mathlib/CategoryTheory/Category/GaloisConnection.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Adjunction.Basic public import Mathlib.Order.GaloisConnection.Defs /-! - # Galois connections between preorders are adjunctions. * `GaloisConnection.adjunction` is the adjunction associated to a Galois connection. diff --git a/Mathlib/CategoryTheory/Category/Preorder.lean b/Mathlib/CategoryTheory/Category/Preorder.lean index 4f84c09560328b..c5d8cc5f61d6f4 100644 --- a/Mathlib/CategoryTheory/Category/Preorder.lean +++ b/Mathlib/CategoryTheory/Category/Preorder.lean @@ -10,7 +10,6 @@ public import Mathlib.Order.Hom.Basic public import Mathlib.Data.ULift /-! - # Preorders as categories We install a category instance on any preorder. This is not to be confused with the category _of_ diff --git a/Mathlib/CategoryTheory/ConcreteCategory/Representable.lean b/Mathlib/CategoryTheory/ConcreteCategory/Representable.lean index fc216bc680d975..a7c113359551b9 100644 --- a/Mathlib/CategoryTheory/ConcreteCategory/Representable.lean +++ b/Mathlib/CategoryTheory/ConcreteCategory/Representable.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Yoneda public import Mathlib.CategoryTheory.ConcreteCategory.Forget /-! - # Representable functors in concrete categories This file provides some API for the situation `(F ⋙ forget D).RepresentableBy Y`. diff --git a/Mathlib/CategoryTheory/Distributive/Cartesian.lean b/Mathlib/CategoryTheory/Distributive/Cartesian.lean index e3db271d2754df..60a080520d0b3e 100644 --- a/Mathlib/CategoryTheory/Distributive/Cartesian.lean +++ b/Mathlib/CategoryTheory/Distributive/Cartesian.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.MonoCoprod public import Mathlib.CategoryTheory.Monoidal.Cartesian.Basic /-! - # Distributive categories ## Main definitions @@ -28,7 +27,6 @@ monoidal distributive. - The coproduct coprojections are monic in a Cartesian distributive category. - ## TODO - Every Cartesian distributive category is finitary distributive, meaning that diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean b/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean index 6b667f8dbe50e3..1ec52a578b70dc 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Basic.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Limits.Shapes.Products /-! - # Effective epimorphisms We define the notion of effective epimorphism and effective epimorphic family of morphisms. diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean b/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean index ddf8c334cc650e..773fd3d33b58fe 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Comp.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.EffectiveEpi.Basic /-! - # Composition of effective epimorphisms This file provides `EffectiveEpi` instances for certain compositions. diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean b/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean index 2e30e726eaeb93..4410e85b583618 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Coproduct.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.EffectiveEpi.Basic public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback public import Mathlib.Tactic.ApplyFun /-! - # Effective epimorphic families and coproducts This file proves that an effective epimorphic family induces an effective epi from the coproduct if diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean b/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean index d4d07cc3c27cfe..672f8aec848533 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Enough.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.EffectiveEpi.Basic /-! - # Effectively enough objects in the image of a functor We define the class `F.EffectivelyEnough` on a functor `F : C ⥤ D` which says that for every object diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean b/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean index 056791f16214a0..010f9dac8313f4 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Extensive.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.EffectiveEpi.Coproduct public import Mathlib.CategoryTheory.Extensive public import Mathlib.CategoryTheory.Limits.Preserves.Finite /-! - # Preserving and reflecting effective epis on extensive categories We prove that a functor between `FinitaryPreExtensive` categories preserves (resp. reflects) finite diff --git a/Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean b/Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean index 22b8d036f7a3f8..b8fd5689da942d 100644 --- a/Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean +++ b/Mathlib/CategoryTheory/EffectiveEpi/Preserves.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.EffectiveEpi.Comp public import Mathlib.CategoryTheory.Limits.Shapes.RegularMono public import Mathlib.CategoryTheory.Limits.Preserves.Basic /-! - # Functors preserving effective epimorphisms This file concerns functors which preserve and/or reflect effective epimorphisms and effective diff --git a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean index 970b1e52b57094..ea0deb1d243308 100644 --- a/Mathlib/CategoryTheory/Endofunctor/Algebra.lean +++ b/Mathlib/CategoryTheory/Endofunctor/Algebra.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal public import Mathlib.CategoryTheory.Functor.EpiMono /-! - # Algebras of endofunctors This file defines (co)algebras of an endofunctor, and provides the category instance for them. diff --git a/Mathlib/CategoryTheory/Enriched/Opposite.lean b/Mathlib/CategoryTheory/Enriched/Opposite.lean index 29c14e67e28b96..24a5efc7401ce8 100644 --- a/Mathlib/CategoryTheory/Enriched/Opposite.lean +++ b/Mathlib/CategoryTheory/Enriched/Opposite.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Enriched.Ordinary.Basic public import Mathlib.CategoryTheory.Monoidal.Braided.Basic /-! - # The opposite category of an enriched category When a monoidal category `V` is braided, we may define the opposite `V`-category of a diff --git a/Mathlib/CategoryTheory/Extensive.lean b/Mathlib/CategoryTheory/Extensive.lean index 3f5988a12aa152..1804c38ac4fc57 100644 --- a/Mathlib/CategoryTheory/Extensive.lean +++ b/Mathlib/CategoryTheory/Extensive.lean @@ -14,7 +14,6 @@ public import Mathlib.CategoryTheory.Limits.MonoCoprod public import Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct /-! - # Extensive categories ## Main definitions diff --git a/Mathlib/CategoryTheory/FiberedCategory/Fiber.lean b/Mathlib/CategoryTheory/FiberedCategory/Fiber.lean index 760a03fef90c0c..dabd1306384715 100644 --- a/Mathlib/CategoryTheory/FiberedCategory/Fiber.lean +++ b/Mathlib/CategoryTheory/FiberedCategory/Fiber.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.FiberedCategory.HomLift public import Mathlib.CategoryTheory.Functor.Const /-! - # Fibers of functors In this file we define, for a functor `p : 𝒳 ⥤ 𝒴`, the fiber categories `Fiber p S` for every diff --git a/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean b/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean index 513d2d7ff5b38d..889ec14581a040 100644 --- a/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean +++ b/Mathlib/CategoryTheory/FiberedCategory/Fibered.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.FiberedCategory.Cartesian /-! - # Fibered categories This file defines what it means for a functor `p : 𝒳 ⥤ 𝒮` to be (pre)fibered. diff --git a/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean b/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean index 18d798ebab7cd3..343d252b69802a 100644 --- a/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean +++ b/Mathlib/CategoryTheory/FiberedCategory/HasFibers.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.FiberedCategory.Fiber public import Mathlib.CategoryTheory.FiberedCategory.Fibered /-! - # Fibers of functors In this file we introduce a typeclass `HasFibers` for a functor `p : 𝒳 ⥤ 𝒮`, consisting of: diff --git a/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean b/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean index 652a48b5d6187d..35f092cf33cd0e 100644 --- a/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean +++ b/Mathlib/CategoryTheory/FiberedCategory/HomLift.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Functor.Category public import Mathlib.CategoryTheory.CommSq /-! - # HomLift Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for expressing the fact that `p(φ) = f` diff --git a/Mathlib/CategoryTheory/Functor/RegularEpi.lean b/Mathlib/CategoryTheory/Functor/RegularEpi.lean index ee15857c11b944..7db2fa1926d8ac 100644 --- a/Mathlib/CategoryTheory/Functor/RegularEpi.lean +++ b/Mathlib/CategoryTheory/Functor/RegularEpi.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Pullbacks public import Mathlib.CategoryTheory.Limits.Shapes.RegularMono /-! - # The category of type-valued sheaves is a regular epi category This file proves that when the target category `D` is a regular epi category (i.e. every epimorphism diff --git a/Mathlib/CategoryTheory/Galois/Action.lean b/Mathlib/CategoryTheory/Galois/Action.lean index 7984a7fab63ff5..85e4f67647140f 100644 --- a/Mathlib/CategoryTheory/Galois/Action.lean +++ b/Mathlib/CategoryTheory/Galois/Action.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Galois.Examples public import Mathlib.CategoryTheory.Galois.Prorepresentability /-! - # Induced functor to finite `Aut F`-sets Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor diff --git a/Mathlib/CategoryTheory/Galois/EssSurj.lean b/Mathlib/CategoryTheory/Galois/EssSurj.lean index 8ee8df169abd5d..514b068af6de5a 100644 --- a/Mathlib/CategoryTheory/Galois/EssSurj.lean +++ b/Mathlib/CategoryTheory/Galois/EssSurj.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Galois.Topology public import Mathlib.Topology.Algebra.OpenSubgroup /-! - # Essential surjectivity of fiber functors Let `F : C ⥤ FintypeCat` be a fiber functor of a Galois category `C` and denote by diff --git a/Mathlib/CategoryTheory/Galois/Full.lean b/Mathlib/CategoryTheory/Galois/Full.lean index e63bb1a322bde5..46daefe1c819ec 100644 --- a/Mathlib/CategoryTheory/Galois/Full.lean +++ b/Mathlib/CategoryTheory/Galois/Full.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Galois.Action /-! - # Fiber functors are (faithfully) full Any (fiber) functor `F : C ⥤ FintypeCat` factors via the forgetful functor diff --git a/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean b/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean index 5cdcfb7e782300..abf716c9085785 100644 --- a/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean +++ b/Mathlib/CategoryTheory/Galois/IsFundamentalgroup.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Galois.Prorepresentability public import Mathlib.Topology.Algebra.OpenSubgroup /-! - # Universal property of fundamental group Let `C` be a Galois category with fiber functor `F`. While in informal mathematics, we tend to diff --git a/Mathlib/CategoryTheory/Galois/Prorepresentability.lean b/Mathlib/CategoryTheory/Galois/Prorepresentability.lean index d33c328d858baa..dc19b58ef4ec8a 100644 --- a/Mathlib/CategoryTheory/Galois/Prorepresentability.lean +++ b/Mathlib/CategoryTheory/Galois/Prorepresentability.lean @@ -263,7 +263,6 @@ theorem AutGalois.π_surjective (A : PointedGaloisObject F) : section EndAutGaloisIsomorphism /-! - ### Isomorphism between `Aut F` and `AutGalois F` In this section we establish the isomorphism between the automorphism group of `F` and diff --git a/Mathlib/CategoryTheory/Galois/Topology.lean b/Mathlib/CategoryTheory/Galois/Topology.lean index b3075869aaa53e..50b088a1e8f361 100644 --- a/Mathlib/CategoryTheory/Galois/Topology.lean +++ b/Mathlib/CategoryTheory/Galois/Topology.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Algebra.ContinuousMonoidHom public import Mathlib.Topology.Algebra.Group.Basic /-! - # Topology of fundamental group In this file we define a natural topology on the automorphism group of a functor diff --git a/Mathlib/CategoryTheory/Groupoid/Discrete.lean b/Mathlib/CategoryTheory/Groupoid/Discrete.lean index e0f1e4843df7fa..e2b312ae53ac6a 100644 --- a/Mathlib/CategoryTheory/Groupoid/Discrete.lean +++ b/Mathlib/CategoryTheory/Groupoid/Discrete.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Groupoid public import Mathlib.CategoryTheory.Discrete.Basic /-! - # Discrete categories are groupoids -/ diff --git a/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean b/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean index d27f71fc9cc941..2ed8237fa816d1 100644 --- a/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean +++ b/Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean @@ -23,7 +23,6 @@ Also defined are: * the notion of normality of subgroupoids and its stability under intersection and preimage; * compatibility of the above with `CategoryTheory.Groupoid.vertexGroup`. - ## Main definitions Given a type `C` with associated `groupoid C` instance. diff --git a/Mathlib/CategoryTheory/Idempotents/Biproducts.lean b/Mathlib/CategoryTheory/Idempotents/Biproducts.lean index e61e76a5cf19b8..1d78c7504c39ff 100644 --- a/Mathlib/CategoryTheory/Idempotents/Biproducts.lean +++ b/Mathlib/CategoryTheory/Idempotents/Biproducts.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Idempotents.Karoubi /-! - # Biproducts in the idempotent completion of a preadditive category In this file, we define an instance expressing that if `C` is an additive category diff --git a/Mathlib/CategoryTheory/Idempotents/SimplicialObject.lean b/Mathlib/CategoryTheory/Idempotents/SimplicialObject.lean index 519d9d4f3d5dfa..5e6130ab1c9e96 100644 --- a/Mathlib/CategoryTheory/Idempotents/SimplicialObject.lean +++ b/Mathlib/CategoryTheory/Idempotents/SimplicialObject.lean @@ -9,7 +9,6 @@ public import Mathlib.AlgebraicTopology.SimplicialObject.Basic public import Mathlib.CategoryTheory.Idempotents.FunctorCategories /-! - # Idempotent completeness of categories of simplicial objects In this file, we provide an instance expressing that `SimplicialObject C` diff --git a/Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean b/Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean index b9ea0cdb10db0d..d6370296ec506d 100644 --- a/Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean +++ b/Mathlib/CategoryTheory/LiftingProperties/Adjunction.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.LiftingProperties.Basic public import Mathlib.CategoryTheory.Adjunction.Basic /-! - # Lifting properties and adjunction In this file, we obtain `Adjunction.HasLiftingProperty_iff`, which states diff --git a/Mathlib/CategoryTheory/Limits/EpiMono.lean b/Mathlib/CategoryTheory/Limits/EpiMono.lean index 49c0441a8cef78..c61c3c4e13eb55 100644 --- a/Mathlib/CategoryTheory/Limits/EpiMono.lean +++ b/Mathlib/CategoryTheory/Limits/EpiMono.lean @@ -25,7 +25,6 @@ X ⟶ Y is a pullback square. - -/ public section diff --git a/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean b/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean index 16ab23d762f949..f7a369378810fa 100644 --- a/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean +++ b/Mathlib/CategoryTheory/Limits/Final/ParallelPair.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Limits.Final /-! - # Conditions for `parallelPair` to be initial In this file we give sufficient conditions on a category `C` and parallel morphisms `f g : X ⟶ Y` diff --git a/Mathlib/CategoryTheory/Limits/FunctorCategory/Finite.lean b/Mathlib/CategoryTheory/Limits/FunctorCategory/Finite.lean index 3a17aa2f0d9c1b..a7a6d8f8f9b797 100644 --- a/Mathlib/CategoryTheory/Limits/FunctorCategory/Finite.lean +++ b/Mathlib/CategoryTheory/Limits/FunctorCategory/Finite.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts public import Mathlib.CategoryTheory.Limits.Preserves.Finite /-! - # Functor categories have finite limits when the target category does These declarations cannot be in `Mathlib/CategoryTheory/Limits/FunctorCategory/Basic.lean` because diff --git a/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Images.lean b/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Images.lean index 395dc3c8619b4a..ca60a23e0c648b 100644 --- a/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Images.lean +++ b/Mathlib/CategoryTheory/Limits/FunctorCategory/Shapes/Images.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Subfunctor.Image public import Mathlib.Tactic.CategoryTheory.CategoryStar /-! - # The category of type-valued functors has images -/ diff --git a/Mathlib/CategoryTheory/Limits/IndYoneda.lean b/Mathlib/CategoryTheory/Limits/IndYoneda.lean index 62e7486874499a..938abb9ece2980 100644 --- a/Mathlib/CategoryTheory/Limits/IndYoneda.lean +++ b/Mathlib/CategoryTheory/Limits/IndYoneda.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.Preserves.Limits public import Mathlib.CategoryTheory.Limits.Yoneda /-! - # Ind- and pro- (co)yoneda lemmas We define limit versions of the yoneda and coyoneda lemmas. diff --git a/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean b/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean index 6e50564c2e8a72..fc875f24687744 100644 --- a/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean +++ b/Mathlib/CategoryTheory/Limits/Indization/ParallelPair.lean @@ -15,7 +15,6 @@ We show that if `A` and `B` are ind-objects and `f` and `g` are natural transfor `A` and `B`, then there is a small filtered category `I` such that `A`, `B`, `f` and `g` are commonly presented by diagrams and natural transformations in `I ⥤ C`. - ## References * [M. Kashiwara, P. Schapira, *Categories and Sheaves*][Kashiwara2006], Proposition 6.1.15 (though our proof is more direct). diff --git a/Mathlib/CategoryTheory/Limits/MonoCoprod.lean b/Mathlib/CategoryTheory/Limits/MonoCoprod.lean index 8f6fd6777fbe73..1ea604d90c02d8 100644 --- a/Mathlib/CategoryTheory/Limits/MonoCoprod.lean +++ b/Mathlib/CategoryTheory/Limits/MonoCoprod.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.RegularMono public import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms /-! - # Categories where inclusions into coproducts are monomorphisms If `C` is a category, the class `MonoCoprod C` expresses that left diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Over.lean b/Mathlib/CategoryTheory/Limits/Preserves/Over.lean index 824f27713a9e47..f6386d9868e958 100644 --- a/Mathlib/CategoryTheory/Limits/Preserves/Over.lean +++ b/Mathlib/CategoryTheory/Limits/Preserves/Over.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Limits.Preserves.Filtered /-! - # Forgetful functor from `Over X` preserves cofiltered limits Note that `Over.forget X : Over X ⥤ C` already preserves all colimits because it is a left adjoint. diff --git a/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean b/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean index 79b3e9d0bc4b7a..f959ef565cb0db 100644 --- a/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean +++ b/Mathlib/CategoryTheory/Limits/Preserves/Ulift.lean @@ -13,7 +13,6 @@ public import Mathlib.Data.Set.Subsingleton /-! # `ULift` creates small (co)limits - This file shows that `uliftFunctor.{v, u}` preserves all limits and colimits, including those potentially too big to exist in `Type u`. diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Connected.lean b/Mathlib/CategoryTheory/Limits/Shapes/Connected.lean index c246e673ef45f3..282b782a463e1d 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/Connected.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/Connected.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.IsConnected public import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks /-! - # Connected shapes In this file we prove that various shapes are connected. diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean b/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean index 4ea64dd9d69f3e..fc30412a2b65a8 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.Equalizers public import Mathlib.CategoryTheory.Limits.ConeCategory /-! - # Multi-(co)equalizers A *multiequalizer* is an equalizer of two morphisms between two products. diff --git a/Mathlib/CategoryTheory/Limits/Shapes/PiProd.lean b/Mathlib/CategoryTheory/Limits/Shapes/PiProd.lean index 67733ba9429866..61e93d0817cef5 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/PiProd.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/PiProd.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts public import Mathlib.CategoryTheory.Limits.Shapes.Products /-! - # A product as a binary product We write a product indexed by `I` as a binary product of the products indexed by a subset of `I` diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Connected.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Connected.lean index dfd33cb4c4fa2e..5c3571e59f1c04 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Connected.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Connected.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.IsConnected public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Defs /-! - # Pullbacks commute with connected limits -/ diff --git a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equifibered.lean b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equifibered.lean index f0d2cfc2ade722..71c6cfa5ba6ba8 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equifibered.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/Pullback/Equifibered.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic public import Mathlib.CategoryTheory.MorphismProperty.Composition /-! - # Equifibered natural transformation ## Main definition diff --git a/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean b/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean index 260c995a2118c9..258f83d890aff1 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/RegularMono.lean @@ -133,7 +133,6 @@ def Fork.IsLimit.regularMono {A B : C} {p₁ p₂ : A ⟶ B} {c : Fork p₁ p₂ section IsRegularMono /-! - Given a regular monomorphism `f : X ⟶ Y` (i.e. a morphism satisfying the predicate `IsRegularMono`), this section gives an equalizer diagram ``` @@ -389,7 +388,6 @@ def Cofork.IsColimit.regularEpi {A B : C} {p₁ p₂ : A ⟶ B} {c : Cofork p₁ section IsRegularEpi /-! - Given a regular epimorphism `f : X ⟶ Y` (i.e. a morphism satisfying the predicate `IsRegularEpi`), this section gives a coequalizer diagram ``` diff --git a/Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean b/Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean index ec42ce0eba4518..76855b0c35578b 100644 --- a/Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean +++ b/Mathlib/CategoryTheory/Limits/Shapes/SequentialProduct.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.PiProd public import Mathlib.Order.Interval.Finset.Nat public import Mathlib.CategoryTheory.EffectiveEpi.Basic /-! - # ℕ-indexed products as sequential limits Given sequences `M N : ℕ → C` of objects with morphisms `f n : M n ⟶ N n` for all `n`, this file diff --git a/Mathlib/CategoryTheory/Limits/VanKampen.lean b/Mathlib/CategoryTheory/Limits/VanKampen.lean index 15d6a166144b66..bee96a20ad040b 100644 --- a/Mathlib/CategoryTheory/Limits/VanKampen.lean +++ b/Mathlib/CategoryTheory/Limits/VanKampen.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial public import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts /-! - # Universal colimits and van Kampen colimits ## Main definitions diff --git a/Mathlib/CategoryTheory/Localization/Construction.lean b/Mathlib/CategoryTheory/Localization/Construction.lean index 25bf8f4eb82a31..091089d49fad5b 100644 --- a/Mathlib/CategoryTheory/Localization/Construction.lean +++ b/Mathlib/CategoryTheory/Localization/Construction.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy public import Mathlib.CategoryTheory.Category.Quiv /-! - # Construction of the localized category This file constructs the localized category, obtained by formally inverting diff --git a/Mathlib/CategoryTheory/Localization/Monoidal/Braided.lean b/Mathlib/CategoryTheory/Localization/Monoidal/Braided.lean index 44e1a7c75839e1..e9359509c197c8 100644 --- a/Mathlib/CategoryTheory/Localization/Monoidal/Braided.lean +++ b/Mathlib/CategoryTheory/Localization/Monoidal/Braided.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Localization.Monoidal.Basic public import Mathlib.CategoryTheory.Monoidal.Braided.Multifunctor /-! - # Localization of symmetric monoidal categories Let `C` be a monoidal category equipped with a class of morphisms `W` which diff --git a/Mathlib/CategoryTheory/Localization/Monoidal/Functor.lean b/Mathlib/CategoryTheory/Localization/Monoidal/Functor.lean index 5fa9970fc17379..e84445eaf4c9a4 100644 --- a/Mathlib/CategoryTheory/Localization/Monoidal/Functor.lean +++ b/Mathlib/CategoryTheory/Localization/Monoidal/Functor.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Monoidal.NaturalTransformation public import Mathlib.Tactic.CategoryTheory.Coherence /-! - # Universal property of localized monoidal categories This file proves that, given a monoidal localization functor `L : C ⥤ D`, and a functor diff --git a/Mathlib/CategoryTheory/Localization/Opposite.lean b/Mathlib/CategoryTheory/Localization/Opposite.lean index 462ce42060bb2d..5d31f2675cd903 100644 --- a/Mathlib/CategoryTheory/Localization/Opposite.lean +++ b/Mathlib/CategoryTheory/Localization/Opposite.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Localization.Equivalence /-! - # Localization of the opposite category If a functor `L : C ⥤ D` is a localization functor for `W : MorphismProperty C`, it diff --git a/Mathlib/CategoryTheory/Localization/Predicate.lean b/Mathlib/CategoryTheory/Localization/Predicate.lean index c882cb52d75bcb..b8504f34fad2c7 100644 --- a/Mathlib/CategoryTheory/Localization/Predicate.lean +++ b/Mathlib/CategoryTheory/Localization/Predicate.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Localization.Construction /-! - # Predicate for localized categories In this file, a predicate `L.IsLocalization W` is introduced for a functor `L : C ⥤ D` diff --git a/Mathlib/CategoryTheory/Monad/EquivMon.lean b/Mathlib/CategoryTheory/Monad/EquivMon.lean index 3152297d85940c..1ffea8a974a7a1 100644 --- a/Mathlib/CategoryTheory/Monad/EquivMon.lean +++ b/Mathlib/CategoryTheory/Monad/EquivMon.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Monoidal.End public import Mathlib.CategoryTheory.Monoidal.Mon /-! - # The equivalence between `Monad C` and `Mon (C ⥤ C)`. A monad "is just" a monoid in the category of endofunctors. diff --git a/Mathlib/CategoryTheory/Monad/Types.lean b/Mathlib/CategoryTheory/Monad/Types.lean index 07cce2be38ad82..16e62c9d549bc7 100644 --- a/Mathlib/CategoryTheory/Monad/Types.lean +++ b/Mathlib/CategoryTheory/Monad/Types.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Types.Basic public import Mathlib.Control.Basic /-! - # Convert from `Monad` (i.e. Lean's `Type`-based monads) to `CategoryTheory.Monad` This allows us to use these monads in category theory. diff --git a/Mathlib/CategoryTheory/Monoidal/Action/Basic.lean b/Mathlib/CategoryTheory/Monoidal/Action/Basic.lean index 6b0534d4b0818e..5310fe9e352742 100644 --- a/Mathlib/CategoryTheory/Monoidal/Action/Basic.lean +++ b/Mathlib/CategoryTheory/Monoidal/Action/Basic.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Monoidal.Category /-! - # Actions from a monoidal category on a category Given a monoidal category `C`, and a category `D`, we define a left action of @@ -21,7 +20,6 @@ We also define right actions, for these, the notation for the action of `c` on `d` is `d ⊙ᵣ c`, and the structure isomorphisms are of the form `- ⊙ᵣ (- ⊗ -) ≅ (- ⊙ᵣ -) ⊙ᵣ -` and `- ⊙ₗ 𝟙_ C ≅ -`. - ## References * [Janelidze, G, and Kelly, G.M., *A note on actions of a monoidal category*][JanelidzeKelly2001] diff --git a/Mathlib/CategoryTheory/Monoidal/Action/Opposites.lean b/Mathlib/CategoryTheory/Monoidal/Action/Opposites.lean index ffd1a2eb940b39..1c2092d3ea5141 100644 --- a/Mathlib/CategoryTheory/Monoidal/Action/Opposites.lean +++ b/Mathlib/CategoryTheory/Monoidal/Action/Opposites.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Monoidal.Action.Basic public import Mathlib.CategoryTheory.Monoidal.Opposite /-! - # Actions from the monoidal opposite of a category. In this file, given a monoidal category `C` and a category `D`, diff --git a/Mathlib/CategoryTheory/Monoidal/Braided/Multifunctor.lean b/Mathlib/CategoryTheory/Monoidal/Braided/Multifunctor.lean index 0b8ec4dbf448c4..3983bd9a0f8bd7 100644 --- a/Mathlib/CategoryTheory/Monoidal/Braided/Multifunctor.lean +++ b/Mathlib/CategoryTheory/Monoidal/Braided/Multifunctor.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Monoidal.Braided.Basic public import Mathlib.CategoryTheory.Functor.CurryingThree /-! - # Constructing braided categories from natural transformations between multifunctors This file provides an alternative constructor for braided categories, given a braiding @@ -97,7 +96,6 @@ example : (bifunctorComp₁₂ (curriedTensor C) (curriedTensor C)) = namespace Forward /-! - ### The forward hexagon identity Given a braiding in the form of a natural isomorphism of bifunctors @@ -153,7 +151,6 @@ end Forward namespace Reverse /-! - ### The reverse hexagon identity Given a braiding in the form of a natural isomorphism of bifunctors diff --git a/Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean b/Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean index b7501b5b3e9eae..8fb4fc21a5cc6a 100644 --- a/Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean +++ b/Mathlib/CategoryTheory/Monoidal/Braided/Reflection.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Monad.Adjunction public import Mathlib.CategoryTheory.Monoidal.Braided.Basic public import Mathlib.Tactic.TFAE /-! - # Day's reflection theorem Let `D` be a symmetric monoidal closed category and let `C` be a reflective subcategory. Day's diff --git a/Mathlib/CategoryTheory/Monoidal/Braided/Transport.lean b/Mathlib/CategoryTheory/Monoidal/Braided/Transport.lean index 14bd87499f12c6..67588d1d9850c4 100644 --- a/Mathlib/CategoryTheory/Monoidal/Braided/Transport.lean +++ b/Mathlib/CategoryTheory/Monoidal/Braided/Transport.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Monoidal.Braided.Basic public import Mathlib.CategoryTheory.Monoidal.Transport /-! - # Transport a symmetric monoidal structure along an equivalence of categories -/ diff --git a/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean b/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean index aa334a5b7de085..5100e84a2c6b94 100644 --- a/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean +++ b/Mathlib/CategoryTheory/Monoidal/Cartesian/Over.lean @@ -13,7 +13,6 @@ public import Mathlib.CategoryTheory.Monoidal.Grp public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic /-! - # `CartesianMonoidalCategory` for `Over X` We provide a `CartesianMonoidalCategory (Over X)` instance via pullbacks, and provide simp lemmas diff --git a/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Complete.lean b/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Complete.lean index 31f5754c824cca..922e56edf28857 100644 --- a/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Complete.lean +++ b/Mathlib/CategoryTheory/Monoidal/Closed/FunctorCategory/Complete.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Groupoid.Discrete public import Mathlib.CategoryTheory.Limits.Preserves.FunctorCategory public import Mathlib.CategoryTheory.Monad.Comonadicity /-! - # Functors into a complete monoidal closed category form a monoidal closed category. TODO (in progress by Joël Riou): make a more explicit construction of the internal hom in functor diff --git a/Mathlib/CategoryTheory/Monoidal/Closed/Transport.lean b/Mathlib/CategoryTheory/Monoidal/Closed/Transport.lean index a45555626bccb4..c6450d250bb4b5 100644 --- a/Mathlib/CategoryTheory/Monoidal/Closed/Transport.lean +++ b/Mathlib/CategoryTheory/Monoidal/Closed/Transport.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Monoidal.Closed.Basic public import Mathlib.CategoryTheory.Monoidal.Transport /-! - # Transporting a closed monoidal structure along an equivalence of categories -/ diff --git a/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean b/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean index a1f3dfdb59c13e..0ae3eec480a5d3 100644 --- a/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean +++ b/Mathlib/CategoryTheory/Monoidal/DayConvolution.lean @@ -33,7 +33,6 @@ the latter were to have the Day convolution monoidal structure. The definition assumptions on `V`) a `LawfulDayConvolutionMonoidalCategoryStruct C V D` to a monoidal structure. - ## References - [nLab page: Day convolution](https://ncatlab.org/nlab/show/Day+convolution) diff --git a/Mathlib/CategoryTheory/Monoidal/Hopf_.lean b/Mathlib/CategoryTheory/Monoidal/Hopf_.lean index 90d793bd1f497f..556c112ea7bd26 100644 --- a/Mathlib/CategoryTheory/Monoidal/Hopf_.lean +++ b/Mathlib/CategoryTheory/Monoidal/Hopf_.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Monoidal.Conv /-! # The category of Hopf monoids in a braided monoidal category. - ## TODO * Show that in a Cartesian monoidal category Hopf monoids are exactly group objects. diff --git a/Mathlib/CategoryTheory/Monoidal/Multifunctor.lean b/Mathlib/CategoryTheory/Monoidal/Multifunctor.lean index 1b3f4711c10904..7e70d4c71decc5 100644 --- a/Mathlib/CategoryTheory/Monoidal/Multifunctor.lean +++ b/Mathlib/CategoryTheory/Monoidal/Multifunctor.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Monoidal.Functor /-! - # Constructing monoidal functors from natural transformations between multifunctors This file provides alternative constructors for (op/lax) monoidal functors, given tensorators @@ -102,7 +101,6 @@ open MonoidalCategory namespace Functor.LaxMonoidal /-! - ## Lax monoidal functors Given a unit morphism `ε : 𝟙_ D ⟶ F.obj (𝟙_ C))` and a tensorator `μ : F - ⊗ F - ⟶ F (- ⊗ -)` @@ -289,7 +287,6 @@ end LaxMonoidal namespace OplaxMonoidal /-! - ## Oplax monoidal functors Given a counit morphism `η : F.obj (𝟙_ C)) ⟶ 𝟙_ D` and a tensorator `δ : F (- ⊗ -) ⟶ F - ⊗ F -` diff --git a/Mathlib/CategoryTheory/MorphismProperty/CommaSites.lean b/Mathlib/CategoryTheory/MorphismProperty/CommaSites.lean index 6a5196bfc86997..8274843a3b9b72 100644 --- a/Mathlib/CategoryTheory/MorphismProperty/CommaSites.lean +++ b/Mathlib/CategoryTheory/MorphismProperty/CommaSites.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.MorphismProperty public import Mathlib.CategoryTheory.Sites.Over /-! - # Sites on `P.Over ⊤ X` We provide some API for proving properties of `P.Over ⊤ X` in relation to precoverages. Consider diff --git a/Mathlib/CategoryTheory/MorphismProperty/Representable.lean b/Mathlib/CategoryTheory/MorphismProperty/Representable.lean index 72a06cb7ab00ec..8802c0eef0dc3c 100644 --- a/Mathlib/CategoryTheory/MorphismProperty/Representable.lean +++ b/Mathlib/CategoryTheory/MorphismProperty/Representable.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.MorphismProperty.Limits /-! - # Relatively representable morphisms In this file we define and develop basic results about relatively representable morphisms. diff --git a/Mathlib/CategoryTheory/ObjectProperty/LimitsOfShape.lean b/Mathlib/CategoryTheory/ObjectProperty/LimitsOfShape.lean index 757761c8d2c7cf..03c13fd0e59a8d 100644 --- a/Mathlib/CategoryTheory/ObjectProperty/LimitsOfShape.lean +++ b/Mathlib/CategoryTheory/ObjectProperty/LimitsOfShape.lean @@ -29,7 +29,6 @@ is essentially small. By requiring `P.limitsOfShape J ≤ P`, we introduce a typeclass `P.IsClosedUnderLimitsOfShape J`. - ## TODO * formalize the closure of `P` under finite limits (which require diff --git a/Mathlib/CategoryTheory/Preadditive/Projective/Internal.lean b/Mathlib/CategoryTheory/Preadditive/Projective/Internal.lean index ee04c3c01d315f..2991d6f3c2d811 100644 --- a/Mathlib/CategoryTheory/Preadditive/Projective/Internal.lean +++ b/Mathlib/CategoryTheory/Preadditive/Projective/Internal.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Monoidal.Closed.Basic public import Mathlib.CategoryTheory.ObjectProperty.Retract /-! - # Internal projectivity This file defines internal projectivity of objects `P` in a category `C` as a class diff --git a/Mathlib/CategoryTheory/Profunctor/Basic.lean b/Mathlib/CategoryTheory/Profunctor/Basic.lean index cf4cca5e5b8fbf..e966fd1d831bc4 100644 --- a/Mathlib/CategoryTheory/Profunctor/Basic.lean +++ b/Mathlib/CategoryTheory/Profunctor/Basic.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Yoneda /-! - # Profunctors A profunctor from a category `C` to a category `D` is a functor from `C` to a category of diff --git a/Mathlib/CategoryTheory/Sites/Adjunction.lean b/Mathlib/CategoryTheory/Sites/Adjunction.lean index a325ec54ea1d9a..92ce58aced6e56 100644 --- a/Mathlib/CategoryTheory/Sites/Adjunction.lean +++ b/Mathlib/CategoryTheory/Sites/Adjunction.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Adjunction.Whiskering public import Mathlib.CategoryTheory.Sites.PreservesSheafification /-! - In this file, we show that an adjunction `G ⊣ F` induces an adjunction between categories of sheaves. We also show that `G` preserves sheafification. diff --git a/Mathlib/CategoryTheory/Sites/CartesianClosed.lean b/Mathlib/CategoryTheory/Sites/CartesianClosed.lean index 6d1e5cc0524653..2261f4e4e1bc32 100644 --- a/Mathlib/CategoryTheory/Sites/CartesianClosed.lean +++ b/Mathlib/CategoryTheory/Sites/CartesianClosed.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Monoidal.Cartesian.FunctorCategory public import Mathlib.CategoryTheory.Sites.CartesianMonoidal public import Mathlib.CategoryTheory.Sites.Sheafification /-! - # Sheaf categories are Cartesian closed ...if the underlying presheaf category is Cartesian closed, the target category has diff --git a/Mathlib/CategoryTheory/Sites/Closed.lean b/Mathlib/CategoryTheory/Sites/Closed.lean index 7ae033f8ffa550..75af6c1280a24f 100644 --- a/Mathlib/CategoryTheory/Sites/Closed.lean +++ b/Mathlib/CategoryTheory/Sites/Closed.lean @@ -37,7 +37,6 @@ that natural closure operators are in bijection with Grothendieck topologies. set of sieves on every object which commutes with pullback additionally induces a Grothendieck topology, giving a bijection with `CategoryTheory.GrothendieckTopology.closureOperator`. - ## Tags closed sieve, closure, Grothendieck topology diff --git a/Mathlib/CategoryTheory/Sites/Coherent/Basic.lean b/Mathlib/CategoryTheory/Sites/Coherent/Basic.lean index fb34080b087687..bb730925cc8c97 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/Basic.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Extensive public import Mathlib.CategoryTheory.Sites.Coverage public import Mathlib.CategoryTheory.EffectiveEpi.Basic /-! - # The Coherent, Regular and Extensive Grothendieck Topologies This file defines three related Grothendieck topologies on a category `C`. diff --git a/Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean index d277c096031618..aead9e88e425ff 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Canonical public import Mathlib.CategoryTheory.Sites.Coherent.Basic public import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic /-! - # Sheaves for the coherent topology This file characterises sheaves for the coherent topology diff --git a/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean b/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean index b09b1098ea1c02..535d6dc2551fb4 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/Comparison.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.Basic public import Mathlib.CategoryTheory.EffectiveEpi.Comp public import Mathlib.CategoryTheory.EffectiveEpi.Extensive /-! - # Connections between the regular, extensive and coherent topologies This file compares the regular, extensive and coherent topologies. diff --git a/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean b/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean index bae4ed2f8cb841..34aab0ec129252 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison public import Mathlib.CategoryTheory.Sites.Equivalence /-! - # Coherence and equivalence of categories This file proves that the coherent and regular topologies transfer nicely along equivalences of diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveColimits.lean b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveColimits.lean index 652e1ed986c063..6a98d45ef4c5d3 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveColimits.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveColimits.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Preadditive.Biproducts public import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves public import Mathlib.CategoryTheory.Sites.Limits /-! - # Colimits in categories of extensive sheaves This file proves that `J`-shaped colimits of `A`-valued sheaves for the extensive topology are diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean index 6e166f73f15ea6..edd17dd8e81c28 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/ExtensiveSheaves.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Sites.Canonical public import Mathlib.CategoryTheory.Sites.Coherent.Basic public import Mathlib.CategoryTheory.Sites.Preserves /-! - # Sheaves for the extensive topology This file characterises sheaves for the extensive topology. diff --git a/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean index 146e646f8df096..f4991e6d7f89e9 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/LocallySurjective.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveTopology public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison public import Mathlib.CategoryTheory.Sites.LocallySurjective /-! - # Locally surjective morphisms of coherent sheaves This file characterises locally surjective morphisms of presheaves for the coherent, regular diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean index bdbb68beb40f4d..0b1efdf8722d52 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPrecoherent.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.EffectiveEpi.Enough public import Mathlib.CategoryTheory.EffectiveEpi.Preserves public import Mathlib.CategoryTheory.Sites.Coherent.CoherentTopology /-! - # Reflecting the property of being precoherent We prove that given a fully faithful functor `F : C ⥤ D` which preserves and reflects finite diff --git a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean index 4abe5bf8e35a88..86be8062c19bc6 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/ReflectsPreregular.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.EffectiveEpi.Enough public import Mathlib.CategoryTheory.EffectiveEpi.Preserves public import Mathlib.CategoryTheory.Sites.Coherent.RegularTopology /-! - # Reflecting the property of being preregular We prove that given a fully faithful functor `F : C ⥤ D`, with `Preregular D`, such that for every diff --git a/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean b/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean index 50b5cccb9a70d0..7603eeef996bb6 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Sites.Canonical public import Mathlib.CategoryTheory.Sites.Coherent.Basic public import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic /-! - # Sheaves for the regular topology This file characterises sheaves for the regular topology. diff --git a/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean b/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean index d971a5b8174760..a9993ee74e58fe 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean @@ -7,7 +7,6 @@ module public import Mathlib.CategoryTheory.Sites.Coherent.RegularSheaves /-! - # Description of the covering sieves of the regular topology This file characterises the covering sieves of the regular topology. diff --git a/Mathlib/CategoryTheory/Sites/Coherent/SequentialLimit.lean b/Mathlib/CategoryTheory/Sites/Coherent/SequentialLimit.lean index 45bf7691b2f641..5f84e03b336c02 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/SequentialLimit.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/SequentialLimit.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective public import Mathlib.CategoryTheory.Sites.EpiMono public import Mathlib.CategoryTheory.Sites.Subcanonical /-! - # Limits of epimorphisms in coherent topoi This file proves that a sequential limit of epimorphisms is epimorphic in the category of sheaves diff --git a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean index 0b2dc999b2717d..ce6a132878caa0 100644 --- a/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean +++ b/Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular public import Mathlib.CategoryTheory.Sites.DenseSubsite.InducedTopology public import Mathlib.CategoryTheory.Sites.Whiskering /-! - # Categories of coherent sheaves Given a fully faithful functor `F : C ⥤ D` into a precoherent category, which preserves and reflects diff --git a/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean b/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean index 51a03ad69b9c5d..585b1b99fcdc47 100644 --- a/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean +++ b/Mathlib/CategoryTheory/Sites/CompatiblePlus.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Whiskering public import Mathlib.CategoryTheory.Sites.Plus /-! - In this file, we prove that the plus functor is compatible with functors which preserve the correct limits and colimits. diff --git a/Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean b/Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean index 5595a48e173681..6affc3aadbe697 100644 --- a/Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean +++ b/Mathlib/CategoryTheory/Sites/CompatibleSheafification.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.CompatiblePlus public import Mathlib.CategoryTheory.Sites.ConcreteSheafification /-! - In this file, we prove that sheafification is compatible with functors which preserve the correct limits and colimits. diff --git a/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean b/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean index e4fea004e81e05..e02bc9c13b37f9 100644 --- a/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean +++ b/Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Plus public import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory /-! - # Sheafification We construct the sheafification of a presheaf over a site `C` with values in `D` whenever diff --git a/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean b/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean index f7b3ccdc91aa3b..ed9a46809bf468 100644 --- a/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean +++ b/Mathlib/CategoryTheory/Sites/ConstantSheaf.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Sheafification public import Mathlib.CategoryTheory.Sites.DenseSubsite.SheafEquiv /-! - # The constant sheaf We define the constant sheaf functor (the sheafification of the constant presheaf) diff --git a/Mathlib/CategoryTheory/Sites/Coverage.lean b/Mathlib/CategoryTheory/Sites/Coverage.lean index 832f1c3ae18087..fe9a8d1f614748 100644 --- a/Mathlib/CategoryTheory/Sites/Coverage.lean +++ b/Mathlib/CategoryTheory/Sites/Coverage.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.PrecoverageToGrothendieck public import Mathlib.CategoryTheory.Sites.Sheaf /-! - # Coverages A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`, diff --git a/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean b/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean index 7f49ecb19f8495..85703384c1a3fa 100644 --- a/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean +++ b/Mathlib/CategoryTheory/Sites/EffectiveEpimorphic.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.SheafOfTypes public import Mathlib.CategoryTheory.EffectiveEpi.Basic /-! - # Effective epimorphic sieves We define the notion of effective epimorphic (pre)sieves and provide some API for relating the diff --git a/Mathlib/CategoryTheory/Sites/Hypercover/SheafOfTypes.lean b/Mathlib/CategoryTheory/Sites/Hypercover/SheafOfTypes.lean index 2ecb59c096232b..61097a9549c0a6 100644 --- a/Mathlib/CategoryTheory/Sites/Hypercover/SheafOfTypes.lean +++ b/Mathlib/CategoryTheory/Sites/Hypercover/SheafOfTypes.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Hypercover.One public import Mathlib.CategoryTheory.Limits.Types.Multiequalizer /-! - # `1`-hypercovers and (pre)sheaves of types In this file we provide some API for working with `1`-hypercovers for sheaves of types. diff --git a/Mathlib/CategoryTheory/Sites/Limits.lean b/Mathlib/CategoryTheory/Sites/Limits.lean index c2b2220ff983ff..b4f13b133bdfe2 100644 --- a/Mathlib/CategoryTheory/Sites/Limits.lean +++ b/Mathlib/CategoryTheory/Sites/Limits.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts public import Mathlib.CategoryTheory.Limits.FullSubcategory /-! - # Limits and colimits of sheaves ## Limits diff --git a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean index c6bcc571e40dc8..ac149c552dd4cb 100644 --- a/Mathlib/CategoryTheory/Sites/LocallySurjective.lean +++ b/Mathlib/CategoryTheory/Sites/LocallySurjective.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Sites.CompatibleSheafification public import Mathlib.CategoryTheory.Sites.LocallyInjective public import Mathlib.CategoryTheory.ShrinkYoneda /-! - # Locally surjective morphisms ## Main definitions diff --git a/Mathlib/CategoryTheory/Sites/Plus.lean b/Mathlib/CategoryTheory/Sites/Plus.lean index c3fd8a143c05ff..708039ab6bdbdf 100644 --- a/Mathlib/CategoryTheory/Sites/Plus.lean +++ b/Mathlib/CategoryTheory/Sites/Plus.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Sheaf /-! - # The plus construction for presheaves. This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D` diff --git a/Mathlib/CategoryTheory/Sites/Precoverage.lean b/Mathlib/CategoryTheory/Sites/Precoverage.lean index 6055d8aa89b92e..aa0ca29342876c 100644 --- a/Mathlib/CategoryTheory/Sites/Precoverage.lean +++ b/Mathlib/CategoryTheory/Sites/Precoverage.lean @@ -11,7 +11,6 @@ public import Mathlib.Order.ConditionallyCompleteLattice.Basic public import Mathlib.CategoryTheory.Limits.Shapes.Pullback.IsPullback.Basic /-! - # Precoverages A precoverage `K` on a category `C` is a set of presieves associated to every object `X : C`, diff --git a/Mathlib/CategoryTheory/Sites/PrecoverageToGrothendieck.lean b/Mathlib/CategoryTheory/Sites/PrecoverageToGrothendieck.lean index fc344b9950c132..18ed80d2c5cd02 100644 --- a/Mathlib/CategoryTheory/Sites/PrecoverageToGrothendieck.lean +++ b/Mathlib/CategoryTheory/Sites/PrecoverageToGrothendieck.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Hypercover.Zero public import Mathlib.CategoryTheory.Sites.SheafOfTypes /-! - # Grothendieck topology generated by a precoverage For any category `C`, we define the Grothendieck topology generated by a precoverage `J` on `C`. diff --git a/Mathlib/CategoryTheory/Sites/PreservesLimits.lean b/Mathlib/CategoryTheory/Sites/PreservesLimits.lean index e4f2844c935df1..a67feacd2cb5dc 100644 --- a/Mathlib/CategoryTheory/Sites/PreservesLimits.lean +++ b/Mathlib/CategoryTheory/Sites/PreservesLimits.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.Sheafification public import Mathlib.CategoryTheory.Limits.Preserves.Finite /-! - # Preservation of (co)limits by the sheaf Yoneda functor -/ diff --git a/Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean b/Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean index 1c5b83f21cbdf5..fb0caf2644f116 100644 --- a/Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean +++ b/Mathlib/CategoryTheory/Sites/PreservesLocallyBijective.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.DenseSubsite.Basic public import Mathlib.CategoryTheory.Sites.LocallySurjective /-! - # Preserving and reflecting local injectivity and surjectivity This file proves that precomposition with a cocontinuous functor preserves local injectivity and diff --git a/Mathlib/CategoryTheory/Sites/RegularEpi.lean b/Mathlib/CategoryTheory/Sites/RegularEpi.lean index 278fc6650fcf23..d7f119213a7f80 100644 --- a/Mathlib/CategoryTheory/Sites/RegularEpi.lean +++ b/Mathlib/CategoryTheory/Sites/RegularEpi.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Limits.FunctorCategory.Shapes.Images public import Mathlib.CategoryTheory.Sites.LeftExact /-! - # The category of type-valued sheaves is a regular epi category ## Main results diff --git a/Mathlib/CategoryTheory/Sites/Sheafification.lean b/Mathlib/CategoryTheory/Sites/Sheafification.lean index 0c9873755a9646..41cb397d890018 100644 --- a/Mathlib/CategoryTheory/Sites/Sheafification.lean +++ b/Mathlib/CategoryTheory/Sites/Sheafification.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Adjunction.Reflective public import Mathlib.CategoryTheory.Sites.Sheaf public import Mathlib.CategoryTheory.Limits.Preserves.Finite /-! - # Sheafification Given a site `(C, J)` we define a typeclass `HasSheafify J A` saying that the inclusion functor from diff --git a/Mathlib/CategoryTheory/Sites/Subcanonical.lean b/Mathlib/CategoryTheory/Sites/Subcanonical.lean index ef05b6fd22bb64..704f1be65f8f1f 100644 --- a/Mathlib/CategoryTheory/Sites/Subcanonical.lean +++ b/Mathlib/CategoryTheory/Sites/Subcanonical.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.Whiskering public import Mathlib.CategoryTheory.Limits.Shapes.DisjointCoproduct public import Mathlib.CategoryTheory.Sites.Continuous /-! - # Subcanonical Grothendieck topologies This file provides some API for the Yoneda embedding into the category of sheaves for a diff --git a/Mathlib/CategoryTheory/Sites/Subsheaf.lean b/Mathlib/CategoryTheory/Sites/Subsheaf.lean index c2b64a3eb952aa..18de0a245b8338 100644 --- a/Mathlib/CategoryTheory/Sites/Subsheaf.lean +++ b/Mathlib/CategoryTheory/Sites/Subsheaf.lean @@ -13,7 +13,6 @@ public import Mathlib.CategoryTheory.Subfunctor.Image public import Mathlib.CategoryTheory.Subfunctor.Sieves /-! - # Subsheaf of types We define the subsheaf of a type-valued presheaf. diff --git a/Mathlib/CategoryTheory/Sites/Whiskering.lean b/Mathlib/CategoryTheory/Sites/Whiskering.lean index bbb916c0ca3f13..9e2e99945a92c5 100644 --- a/Mathlib/CategoryTheory/Sites/Whiskering.lean +++ b/Mathlib/CategoryTheory/Sites/Whiskering.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Sheaf public import Mathlib.CategoryTheory.ConcreteCategory.Forget /-! - In this file we construct the functor `Sheaf J A ⥤ Sheaf J B` between sheaf categories obtained by composition with a functor `F : A ⥤ B`. diff --git a/Mathlib/CategoryTheory/Subfunctor/Basic.lean b/Mathlib/CategoryTheory/Subfunctor/Basic.lean index 741777bea72237..7d6e7fa48fdcfe 100644 --- a/Mathlib/CategoryTheory/Subfunctor/Basic.lean +++ b/Mathlib/CategoryTheory/Subfunctor/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Types.Basic public import Mathlib.Data.Set.Lattice.Image /-! - # Subfunctor of types We define subfunctors of a type-valued functors. diff --git a/Mathlib/CategoryTheory/Subobject/Classifier/Defs.lean b/Mathlib/CategoryTheory/Subobject/Classifier/Defs.lean index bce43342532765..5f9ceaa0ba97ba 100644 --- a/Mathlib/CategoryTheory/Subobject/Classifier/Defs.lean +++ b/Mathlib/CategoryTheory/Subobject/Classifier/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Functor.ReflectsIso.Balanced public import Mathlib.CategoryTheory.Subobject.Presheaf /-! - # Subobject Classifier We define a structure containing the data of a subobject classifier in a category `C` as diff --git a/Mathlib/CategoryTheory/Topos/Sheaf.lean b/Mathlib/CategoryTheory/Topos/Sheaf.lean index 3260f5e1bd332a..aac889df5e0356 100644 --- a/Mathlib/CategoryTheory/Topos/Sheaf.lean +++ b/Mathlib/CategoryTheory/Topos/Sheaf.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Subobject.Classifier.Defs public import Mathlib.CategoryTheory.Subfunctor.Image /-! - # (Elementary) Sheaf Topos We define a subobject classifier for categories of sheaves of (large enough) types. diff --git a/Mathlib/CategoryTheory/WithTerminal/Basic.lean b/Mathlib/CategoryTheory/WithTerminal/Basic.lean index f4986a2e85a677..b00ab6ff5237b5 100644 --- a/Mathlib/CategoryTheory/WithTerminal/Basic.lean +++ b/Mathlib/CategoryTheory/WithTerminal/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks public import Mathlib.CategoryTheory.Bicategory.Functor.Pseudofunctor /-! - # `WithInitial` and `WithTerminal` Given a category `C`, this file constructs two objects: diff --git a/Mathlib/CategoryTheory/WithTerminal/FinCategory.lean b/Mathlib/CategoryTheory/WithTerminal/FinCategory.lean index 69d652bbfc1ff7..37d91cca70ddbd 100644 --- a/Mathlib/CategoryTheory/WithTerminal/FinCategory.lean +++ b/Mathlib/CategoryTheory/WithTerminal/FinCategory.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.WithTerminal.Basic public import Mathlib.Data.Fintype.Option /-! - # `WithTerminal C` and `WithInitial C` are finite whenever `C` is If `C` has finitely many objects, then so do `WithTerminal C` and `WithInitial C`, diff --git a/Mathlib/Combinatorics/Digraph/Orientation.lean b/Mathlib/Combinatorics/Digraph/Orientation.lean index eed173238dc4e7..e95477897bf967 100644 --- a/Mathlib/Combinatorics/Digraph/Orientation.lean +++ b/Mathlib/Combinatorics/Digraph/Orientation.lean @@ -9,7 +9,6 @@ public import Mathlib.Combinatorics.Digraph.Basic public import Mathlib.Combinatorics.SimpleGraph.Basic /-! - # Graph Orientation This module introduces conversion operations between `Digraph`s and `SimpleGraph`s, by forgetting diff --git a/Mathlib/Combinatorics/Matroid/IndepAxioms.lean b/Mathlib/Combinatorics/Matroid/IndepAxioms.lean index e1c915a06ae2e9..a2fca050884a89 100644 --- a/Mathlib/Combinatorics/Matroid/IndepAxioms.lean +++ b/Mathlib/Combinatorics/Matroid/IndepAxioms.lean @@ -53,7 +53,6 @@ for the inverse of `e`). ## Main definitions - * `IndepMatroid α` is a matroid structure on `α` described in terms of its independent sets in full generality, using infinite versions of the axioms. diff --git a/Mathlib/Combinatorics/Optimization/ValuedCSP.lean b/Mathlib/Combinatorics/Optimization/ValuedCSP.lean index 851c04232349f7..f9dd644e9fe459 100644 --- a/Mathlib/Combinatorics/Optimization/ValuedCSP.lean +++ b/Mathlib/Combinatorics/Optimization/ValuedCSP.lean @@ -11,7 +11,6 @@ public import Mathlib.Data.Fin.VecNotation public import Mathlib.LinearAlgebra.Matrix.Notation /-! - # General-Valued Constraint Satisfaction Problems General-Valued CSP is a very broad class of problems in discrete optimization. diff --git a/Mathlib/Combinatorics/Quiver/Cast.lean b/Mathlib/Combinatorics/Quiver/Cast.lean index dd1155365c1c7b..7bbc8b1da246da 100644 --- a/Mathlib/Combinatorics/Quiver/Cast.lean +++ b/Mathlib/Combinatorics/Quiver/Cast.lean @@ -9,7 +9,6 @@ public import Mathlib.Combinatorics.Quiver.Basic public import Mathlib.Combinatorics.Quiver.Path /-! - # Rewriting arrows and paths along vertex equalities This file defines `Hom.cast` and `Path.cast` (and associated lemmas) in order to allow diff --git a/Mathlib/Combinatorics/Quiver/Push.lean b/Mathlib/Combinatorics/Quiver/Push.lean index bce18e536a9aad..a4aece6a860668 100644 --- a/Mathlib/Combinatorics/Quiver/Push.lean +++ b/Mathlib/Combinatorics/Quiver/Push.lean @@ -8,7 +8,6 @@ module public import Mathlib.Combinatorics.Quiver.Prefunctor /-! - # Pushing a quiver structure along a map Given a map `σ : V → W` and a `Quiver` instance on `V`, this file defines a `Quiver` instance diff --git a/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean b/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean index 297353a932b46e..bcceb0226ba26b 100644 --- a/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean +++ b/Mathlib/Combinatorics/SimpleGraph/Acyclic.lean @@ -13,7 +13,6 @@ public import Mathlib.Combinatorics.SimpleGraph.DegreeSum public import Mathlib.Combinatorics.SimpleGraph.Metric /-! - # Acyclic graphs and trees This module introduces *acyclic graphs* (a.k.a. *forests*) and *trees*. diff --git a/Mathlib/Combinatorics/SimpleGraph/Paths.lean b/Mathlib/Combinatorics/SimpleGraph/Paths.lean index ada1da35a5daf0..955c18cd1e82fa 100644 --- a/Mathlib/Combinatorics/SimpleGraph/Paths.lean +++ b/Mathlib/Combinatorics/SimpleGraph/Paths.lean @@ -11,7 +11,6 @@ public import Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks public import Mathlib.Order.Preorder.Finite /-! - # Trail, Path, and Cycle In a simple graph, diff --git a/Mathlib/Combinatorics/SimpleGraph/Trails.lean b/Mathlib/Combinatorics/SimpleGraph/Trails.lean index d2e877fc573e22..044c715b89f343 100644 --- a/Mathlib/Combinatorics/SimpleGraph/Trails.lean +++ b/Mathlib/Combinatorics/SimpleGraph/Trails.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Ring.Parity public import Mathlib.Combinatorics.SimpleGraph.Paths /-! - # Trails and Eulerian trails This module contains additional theory about trails, including Eulerian trails (also known diff --git a/Mathlib/Combinatorics/Young/SemistandardTableau.lean b/Mathlib/Combinatorics/Young/SemistandardTableau.lean index dbf52e01a9f9d3..f08ac842e28efd 100644 --- a/Mathlib/Combinatorics/Young/SemistandardTableau.lean +++ b/Mathlib/Combinatorics/Young/SemistandardTableau.lean @@ -25,7 +25,6 @@ An example of an SSYT of shape `μ = [4, 2, 1]` is: We represent a semistandard Young tableau as a function `ℕ → ℕ → ℕ`, which is required to be zero for all pairs `(i, j) ∉ μ` and to satisfy the row-weak and column-strict conditions on `μ`. - ## Main definitions - `SemistandardYoungTableau (μ : YoungDiagram)`: semistandard Young tableaux of shape `μ`. There is diff --git a/Mathlib/Computability/AkraBazzi/SumTransform.lean b/Mathlib/Computability/AkraBazzi/SumTransform.lean index 227670ace0f31f..40b410d5732184 100644 --- a/Mathlib/Computability/AkraBazzi/SumTransform.lean +++ b/Mathlib/Computability/AkraBazzi/SumTransform.lean @@ -32,7 +32,6 @@ We develop further preliminaries required for the theorem, up to the sum transfo namely `n^p (1 + ∑ g(u) / u^(p+1))`, together with positivity statements along the branches `r i n`. - ## References * Mohamad Akra and Louay Bazzi, On the solution of linear recurrence equations diff --git a/Mathlib/Condensed/AB.lean b/Mathlib/Condensed/AB.lean index 4c731d5cb90051..84ff4c82ce0a1c 100644 --- a/Mathlib/Condensed/AB.lean +++ b/Mathlib/Condensed/AB.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveColimits public import Mathlib.Condensed.Equivalence public import Mathlib.Condensed.Limits /-! - # AB axioms in condensed modules This file proves that the category of condensed modules over a ring satisfies Grothendieck's axioms diff --git a/Mathlib/Condensed/Basic.lean b/Mathlib/Condensed/Basic.lean index 926e02ac2875d9..07e854bedc2ae8 100644 --- a/Mathlib/Condensed/Basic.lean +++ b/Mathlib/Condensed/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Sheaf public import Mathlib.Topology.Category.CompHaus.EffectiveEpi /-! - # Condensed Objects This file defines the category of condensed objects in a category `C`, following the work diff --git a/Mathlib/Condensed/CartesianClosed.lean b/Mathlib/Condensed/CartesianClosed.lean index 0ce243384482c5..ff5ceea31bfa29 100644 --- a/Mathlib/Condensed/CartesianClosed.lean +++ b/Mathlib/Condensed/CartesianClosed.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Sites.CartesianClosed public import Mathlib.Condensed.Basic public import Mathlib.CategoryTheory.Sites.LeftExact /-! - # Condensed sets form a Cartesian closed category -/ diff --git a/Mathlib/Condensed/Discrete/Basic.lean b/Mathlib/Condensed/Discrete/Basic.lean index 064c024b17326b..90a6cf1e9bf27d 100644 --- a/Mathlib/Condensed/Discrete/Basic.lean +++ b/Mathlib/Condensed/Discrete/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Condensed.Basic public import Mathlib.Condensed.Light.Basic public import Mathlib.Condensed.Light.Instances /-! - # Discrete-underlying adjunction Given a category `C` with sheafification with respect to the coherent topology on compact Hausdorff diff --git a/Mathlib/Condensed/Discrete/Characterization.lean b/Mathlib/Condensed/Discrete/Characterization.lean index 4ec71ead5d9179..76fddf8753c82f 100644 --- a/Mathlib/Condensed/Discrete/Characterization.lean +++ b/Mathlib/Condensed/Discrete/Characterization.lean @@ -8,7 +8,6 @@ module public import Mathlib.Condensed.Discrete.Colimit public import Mathlib.Condensed.Discrete.Module /-! - # Characterizing discrete condensed sets and `R`-modules. This file proves a characterization of discrete condensed sets, discrete condensed `R`-modules over diff --git a/Mathlib/Condensed/Discrete/Colimit.lean b/Mathlib/Condensed/Discrete/Colimit.lean index 789a1ec6e5c50d..eacd030b8d0f98 100644 --- a/Mathlib/Condensed/Discrete/Colimit.lean +++ b/Mathlib/Condensed/Discrete/Colimit.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.Equivalence public import Mathlib.Topology.Category.LightProfinite.Extend /-! - # The condensed set given by left Kan extension from `FintypeCat` to `Profinite`. This file provides the necessary API to prove that a condensed set `X` is discrete if and only if diff --git a/Mathlib/Condensed/Discrete/LocallyConstant.lean b/Mathlib/Condensed/Discrete/LocallyConstant.lean index aeadb3ecb9f5d6..9d97a6d123647c 100644 --- a/Mathlib/Condensed/Discrete/LocallyConstant.lean +++ b/Mathlib/Condensed/Discrete/LocallyConstant.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.TopComparison public import Mathlib.Topology.Category.CompHausLike.SigmaComparison public import Mathlib.Topology.FiberPartition /-! - # The sheaf of locally constant maps on `CompHausLike P` This file proves that under suitable conditions, the functor from the category of sets to the diff --git a/Mathlib/Condensed/Discrete/Module.lean b/Mathlib/Condensed/Discrete/Module.lean index b3ba358aa6b01b..8d21a2cbfa51b6 100644 --- a/Mathlib/Condensed/Discrete/Module.lean +++ b/Mathlib/Condensed/Discrete/Module.lean @@ -11,7 +11,6 @@ public import Mathlib.Condensed.Light.Module public import Mathlib.Condensed.Module public import Mathlib.Topology.LocallyConstant.Algebra /-! - # Discrete condensed `R`-modules This file provides the necessary API to prove that a condensed `R`-module is discrete if and only diff --git a/Mathlib/Condensed/EffectiveEpi.lean b/Mathlib/Condensed/EffectiveEpi.lean index 412b91471f9f6c..74a4c6b321493d 100644 --- a/Mathlib/Condensed/EffectiveEpi.lean +++ b/Mathlib/Condensed/EffectiveEpi.lean @@ -11,7 +11,6 @@ public import Mathlib.Condensed.Functors public import Mathlib.Condensed.Limits -- shake: keep (compHausToCondensed.PreservesEffectiveEpis), cf. lean#13417 /-! - # The functor from compact Hausdorff spaces to condensed sets preserves effective epimorphisms -/ diff --git a/Mathlib/Condensed/Epi.lean b/Mathlib/Condensed/Epi.lean index 566eb699008b51..191380153edf0e 100644 --- a/Mathlib/Condensed/Epi.lean +++ b/Mathlib/Condensed/Epi.lean @@ -11,7 +11,6 @@ public import Mathlib.CategoryTheory.Sites.EpiMono public import Mathlib.Condensed.Equivalence public import Mathlib.Condensed.Module /-! - # Epimorphisms of condensed objects This file characterises epimorphisms of condensed sets and condensed `R`-modules for any ring `R`, diff --git a/Mathlib/Condensed/Equivalence.lean b/Mathlib/Condensed/Equivalence.lean index db55dae6edac9b..ce75f8e1490220 100644 --- a/Mathlib/Condensed/Equivalence.lean +++ b/Mathlib/Condensed/Equivalence.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Category.Stonean.EffectiveEpi public import Mathlib.Condensed.Basic public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison /-! - # Sheaves on CompHaus are equivalent to sheaves on Stonean The forgetful functor from extremally disconnected spaces `Stonean` to compact diff --git a/Mathlib/Condensed/Explicit.lean b/Mathlib/Condensed/Explicit.lean index 95b37530040315..4faaa42216ae6b 100644 --- a/Mathlib/Condensed/Explicit.lean +++ b/Mathlib/Condensed/Explicit.lean @@ -8,7 +8,6 @@ module public import Mathlib.Condensed.Module public import Mathlib.Condensed.Equivalence /-! - # The explicit sheaf condition for condensed sets We give the following three explicit descriptions of condensed objects: diff --git a/Mathlib/Condensed/Light/Basic.lean b/Mathlib/Condensed/Light/Basic.lean index 7434e0d09e87b4..5caa707c9d15d3 100644 --- a/Mathlib/Condensed/Light/Basic.lean +++ b/Mathlib/Condensed/Light/Basic.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Sheaf public import Mathlib.Topology.Category.LightProfinite.EffectiveEpi /-! - # Light condensed objects This file defines the category of light condensed objects in a category `C`, following the work diff --git a/Mathlib/Condensed/Light/EffectiveEpi.lean b/Mathlib/Condensed/Light/EffectiveEpi.lean index 64d2cda588c579..331052412a35d2 100644 --- a/Mathlib/Condensed/Light/EffectiveEpi.lean +++ b/Mathlib/Condensed/Light/EffectiveEpi.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.Light.Epi public import Mathlib.Condensed.Light.Functors /-! - # The functor from light profinite sets to light condensed sets preserves effective epimorphisms -/ diff --git a/Mathlib/Condensed/Light/Epi.lean b/Mathlib/Condensed/Light/Epi.lean index a6bd9883808a2f..a2e6f51254b2e3 100644 --- a/Mathlib/Condensed/Light/Epi.lean +++ b/Mathlib/Condensed/Light/Epi.lean @@ -10,7 +10,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.SequentialLimit public import Mathlib.Condensed.Light.Functors public import Mathlib.Condensed.Light.Limits /-! - # Epimorphisms of light condensed objects This file characterises epimorphisms in light condensed sets and modules as the locally surjective diff --git a/Mathlib/Condensed/Light/Explicit.lean b/Mathlib/Condensed/Light/Explicit.lean index 68a86d2ac0f377..1c428e71069219 100644 --- a/Mathlib/Condensed/Light/Explicit.lean +++ b/Mathlib/Condensed/Light/Explicit.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison public import Mathlib.Condensed.Light.Module /-! - # The explicit sheaf condition for light condensed sets We give an explicit description of light condensed sets: diff --git a/Mathlib/Condensed/Light/InternallyProjective.lean b/Mathlib/Condensed/Light/InternallyProjective.lean index cd447dc9d07552..a21dc828633803 100644 --- a/Mathlib/Condensed/Light/InternallyProjective.lean +++ b/Mathlib/Condensed/Light/InternallyProjective.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.Light.Epi public import Mathlib.Condensed.Light.Functors public import Mathlib.Condensed.Light.Monoidal /-! - # Characterization of internal projectivity in light condensed modules This file gives an explicit condition on light condensed modules over a ring `R` to be internally diff --git a/Mathlib/Condensed/Light/Limits.lean b/Mathlib/Condensed/Light/Limits.lean index 8f6e66e8f3f1ef..9ccdb27c701c33 100644 --- a/Mathlib/Condensed/Light/Limits.lean +++ b/Mathlib/Condensed/Light/Limits.lean @@ -7,7 +7,6 @@ module public import Mathlib.Condensed.Light.Module /-! - # Limits in categories of light condensed objects This file adds some instances for limits in light condensed sets and modules. diff --git a/Mathlib/Condensed/Light/Module.lean b/Mathlib/Condensed/Light/Module.lean index 909d0c11b9d799..7ee2ecf961b51e 100644 --- a/Mathlib/Condensed/Light/Module.lean +++ b/Mathlib/Condensed/Light/Module.lean @@ -15,7 +15,6 @@ public import Mathlib.CategoryTheory.Sites.Equivalence public import Mathlib.Condensed.Light.Basic public import Mathlib.Condensed.Light.Instances /-! - # Light condensed `R`-modules This file defines light condensed modules over a ring `R`. diff --git a/Mathlib/Condensed/Light/Monoidal.lean b/Mathlib/Condensed/Light/Monoidal.lean index 88dc2bba7dae31..eb23aeacd41970 100644 --- a/Mathlib/Condensed/Light/Monoidal.lean +++ b/Mathlib/Condensed/Light/Monoidal.lean @@ -17,7 +17,6 @@ public import Mathlib.Condensed.Light.Instances public import Mathlib.Condensed.Light.Module /-! - # Closed symmetric monoidal structure on light condensed modules We define a symmetric monoidal structure on light condensed modules by localizing the symmetric diff --git a/Mathlib/Condensed/Light/Sequence.lean b/Mathlib/Condensed/Light/Sequence.lean index 3b9d812f0b7599..399528e27ad962 100644 --- a/Mathlib/Condensed/Light/Sequence.lean +++ b/Mathlib/Condensed/Light/Sequence.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.Light.InternallyProjective import Mathlib.Condensed.Light.EffectiveEpi import Mathlib.Topology.Category.LightProfinite.Injective /-! - # The free light condensed `R`-module `R[ℕ∪∞]` is internally projective -/ diff --git a/Mathlib/Condensed/Light/Small.lean b/Mathlib/Condensed/Light/Small.lean index d76fa6a788764a..6823c9ada3a1a0 100644 --- a/Mathlib/Condensed/Light/Small.lean +++ b/Mathlib/Condensed/Light/Small.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Equivalence public import Mathlib.Condensed.Light.Module /-! - # Equivalence of light condensed objects with sheaves on a small site -/ diff --git a/Mathlib/Condensed/Light/TopCatAdjunction.lean b/Mathlib/Condensed/Light/TopCatAdjunction.lean index af443af3c6f7c4..5a684a676ed774 100644 --- a/Mathlib/Condensed/Light/TopCatAdjunction.lean +++ b/Mathlib/Condensed/Light/TopCatAdjunction.lean @@ -9,7 +9,6 @@ public import Mathlib.Condensed.Light.TopComparison public import Mathlib.Topology.Category.Sequential public import Mathlib.Topology.Category.LightProfinite.Sequence /-! - # The adjunction between light condensed sets and topological spaces This file defines the functor `lightCondSetToTopCat : LightCondSet.{u} ⥤ TopCat.{u}` which is diff --git a/Mathlib/Condensed/Light/TopComparison.lean b/Mathlib/Condensed/Light/TopComparison.lean index 576cc11b4655e0..4f7570b61913ae 100644 --- a/Mathlib/Condensed/Light/TopComparison.lean +++ b/Mathlib/Condensed/Light/TopComparison.lean @@ -9,7 +9,6 @@ public import Mathlib.Condensed.Light.Basic public import Mathlib.Condensed.TopComparison /-! - # The functor from topological spaces to light condensed sets We define the functor `topCatToLightCondSet : TopCat.{u} ⥤ LightCondSet.{u}`. diff --git a/Mathlib/Condensed/Limits.lean b/Mathlib/Condensed/Limits.lean index f1c0ca716ec0e5..3b0da9084fa819 100644 --- a/Mathlib/Condensed/Limits.lean +++ b/Mathlib/Condensed/Limits.lean @@ -8,7 +8,6 @@ module public import Mathlib.Condensed.Module /-! - # Limits in categories of condensed objects This file adds some instances for limits in condensed sets and condensed modules. diff --git a/Mathlib/Condensed/Module.lean b/Mathlib/Condensed/Module.lean index f5834efa0d5bf1..8d79f07018ba77 100644 --- a/Mathlib/Condensed/Module.lean +++ b/Mathlib/Condensed/Module.lean @@ -14,7 +14,6 @@ public import Mathlib.CategoryTheory.Sites.Adjunction public import Mathlib.CategoryTheory.Sites.LeftExact public import Mathlib.Condensed.Basic /-! - # Condensed `R`-modules This file defines condensed modules over a ring `R`. diff --git a/Mathlib/Condensed/Solid.lean b/Mathlib/Condensed/Solid.lean index f5214433f91ee8..d1b550ae941c2f 100644 --- a/Mathlib/Condensed/Solid.lean +++ b/Mathlib/Condensed/Solid.lean @@ -10,7 +10,6 @@ public import Mathlib.Condensed.Functors public import Mathlib.Condensed.Limits /-! - # Solid modules This file contains the definition of a solid `R`-module: `CondensedMod.isSolid R`. Solid modules diff --git a/Mathlib/Condensed/TopCatAdjunction.lean b/Mathlib/Condensed/TopCatAdjunction.lean index bff73e2c99cbb5..6fb79be56e6f08 100644 --- a/Mathlib/Condensed/TopCatAdjunction.lean +++ b/Mathlib/Condensed/TopCatAdjunction.lean @@ -8,7 +8,6 @@ module public import Mathlib.Condensed.TopComparison public import Mathlib.Topology.Category.CompactlyGenerated /-! - # The adjunction between condensed sets and topological spaces This file defines the functor `condensedSetToTopCat : CondensedSet.{u} ⥤ TopCat.{u + 1}` which is diff --git a/Mathlib/Condensed/TopComparison.lean b/Mathlib/Condensed/TopComparison.lean index 11985db7efee7a..1f277da93a5f20 100644 --- a/Mathlib/Condensed/TopComparison.lean +++ b/Mathlib/Condensed/TopComparison.lean @@ -11,7 +11,6 @@ public import Mathlib.Condensed.Basic public import Mathlib.Topology.Category.TopCat.Yoneda /-! - # The functor from topological spaces to condensed sets This file builds on the API from the file `TopCat.Yoneda`. If the forgetful functor to `TopCat` has diff --git a/Mathlib/Control/Bitraversable/Lemmas.lean b/Mathlib/Control/Bitraversable/Lemmas.lean index d842c3b638e157..626a8903a1243b 100644 --- a/Mathlib/Control/Bitraversable/Lemmas.lean +++ b/Mathlib/Control/Bitraversable/Lemmas.lean @@ -31,7 +31,6 @@ with the applicatives `id` and `comp` traversable bitraversable functor bifunctor applicative - -/ public section diff --git a/Mathlib/Control/Fold.lean b/Mathlib/Control/Fold.lean index 13a306db3fc76f..7ccbc13a4b44c2 100644 --- a/Mathlib/Control/Fold.lean +++ b/Mathlib/Control/Fold.lean @@ -15,7 +15,6 @@ public import Mathlib.Control.Traversable.Lemmas public import Mathlib.Tactic.AdaptationNote /-! - # List folds generalized to `Traversable` Informally, we can think of `foldl` as a special case of `traverse` where we do not care about the diff --git a/Mathlib/Control/Functor/Multivariate.lean b/Mathlib/Control/Functor/Multivariate.lean index 493f39fa6c8857..e290fc172279c5 100644 --- a/Mathlib/Control/Functor/Multivariate.lean +++ b/Mathlib/Control/Functor/Multivariate.lean @@ -10,7 +10,6 @@ public import Mathlib.Data.TypeVec public import Mathlib.Logic.Equiv.Defs /-! - # Functors between the category of tuples of types, and the category Type Features: diff --git a/Mathlib/Control/ULiftable.lean b/Mathlib/Control/ULiftable.lean index 95571f205712d2..6b700aab2c9a75 100644 --- a/Mathlib/Control/ULiftable.lean +++ b/Mathlib/Control/ULiftable.lean @@ -26,7 +26,6 @@ to transport over to `Option.{v}`. `ULiftable` is an attempt at improving the si `Option.{u}` in a context that requires `Option.{v}`. It is often used in tandem with `ULift` but the two are purposefully decoupled. - ## Main definitions * `ULiftable` class diff --git a/Mathlib/Data/ENNReal/Basic.lean b/Mathlib/Data/ENNReal/Basic.lean index 218392836d7fad..9e7b72a5457bf4 100644 --- a/Mathlib/Data/ENNReal/Basic.lean +++ b/Mathlib/Data/ENNReal/Basic.lean @@ -31,7 +31,6 @@ making it into a `DivInvOneMonoid`. As a consequence of being a `DivInvOneMonoid`, `ℝ≥0∞` inherits a power operation with integer exponent: this and other properties is shown in `Data.ENNReal.Inv`. - ## Main definitions * `ℝ≥0∞`: the extended nonnegative real numbers `[0, ∞]`; defined as `WithTop ℝ≥0`; it is diff --git a/Mathlib/Data/Fin/Tuple/Sort.lean b/Mathlib/Data/Fin/Tuple/Sort.lean index fc5b8b090beaa8..d4edba19da12f9 100644 --- a/Mathlib/Data/Fin/Tuple/Sort.lean +++ b/Mathlib/Data/Fin/Tuple/Sort.lean @@ -12,7 +12,6 @@ public import Mathlib.Order.Interval.Finset.Fin public import Mathlib.Data.Fintype.Fin /-! - # Sorting tuples by their values Given an `n`-tuple `f : Fin n → α` where `α` is ordered, diff --git a/Mathlib/Data/Finset/Lattice/Basic.lean b/Mathlib/Data/Finset/Lattice/Basic.lean index 7249c420ec26c7..47bd210a32a081 100644 --- a/Mathlib/Data/Finset/Lattice/Basic.lean +++ b/Mathlib/Data/Finset/Lattice/Basic.lean @@ -22,7 +22,6 @@ In Lean, we use lattice notation to talk about things involving unions and inter `Mathlib/Order/Lattice.lean`. For the lattice structure on finsets, `⊥` is called `bot` with `⊥ = ∅` and `⊤` is called `top` with `⊤ = univ`. - ## Implementation Notes All the theorems and instances expect `DecidableEq` instance for `α` diff --git a/Mathlib/Data/Finsupp/Fintype.lean b/Mathlib/Data/Finsupp/Fintype.lean index 29736c74da1e94..d63eacb085ebe8 100644 --- a/Mathlib/Data/Finsupp/Fintype.lean +++ b/Mathlib/Data/Finsupp/Fintype.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Finsupp.Single public import Mathlib.Data.Fintype.BigOperators /-! - # Finiteness and infiniteness of `Finsupp` Some lemmas on the combination of `Finsupp`, `Fintype` and `Infinite`. diff --git a/Mathlib/Data/Finsupp/Weight.lean b/Mathlib/Data/Finsupp/Weight.lean index e937d3e86043a7..b5acb8db3edbda 100644 --- a/Mathlib/Data/Finsupp/Weight.lean +++ b/Mathlib/Data/Finsupp/Weight.lean @@ -58,7 +58,6 @@ as well as a function `w : σ → M`. (The important case is `R = ℕ`.) - For `Finite σ`, `Finsupp.finite_of_degree_le` proves that there are finitely many `f : σ →₀ ℕ` of bounded degree. - ## TODO * Maybe `Finsupp.weight w` and `Finsupp.degree` should have similar types, diff --git a/Mathlib/Data/Fintype/List.lean b/Mathlib/Data/Fintype/List.lean index 536953e26212ba..6eb7a78efeaa9d 100644 --- a/Mathlib/Data/Fintype/List.lean +++ b/Mathlib/Data/Fintype/List.lean @@ -10,7 +10,6 @@ public import Mathlib.Data.Fintype.Defs public import Mathlib.Data.List.Permutation /-! - # Fintype instance for nodup lists The subtype of `{l : List α // l.Nodup}` over a `[Fintype α]` diff --git a/Mathlib/Data/Int/Fib/Basic.lean b/Mathlib/Data/Int/Fib/Basic.lean index e38a4d23db051a..a889b154daa401 100644 --- a/Mathlib/Data/Int/Fib/Basic.lean +++ b/Mathlib/Data/Int/Fib/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Group.Int.Even public import Mathlib.Data.Nat.Fib.Basic /-! - # Fibonacci numbers extended onto the integers This file defines the Fibonacci sequence on the integers. diff --git a/Mathlib/Data/Int/ModEq.lean b/Mathlib/Data/Int/ModEq.lean index 2af03e37ca02ab..5d2bb5700e6b09 100644 --- a/Mathlib/Data/Int/ModEq.lean +++ b/Mathlib/Data/Int/ModEq.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Nat.ModEq import Mathlib.Data.Int.Cast.Lemmas /-! - # Congruences modulo an integer This file defines the equivalence relation `a ≡ b [ZMOD n]` on the integers, similarly to how diff --git a/Mathlib/Data/List/DropRight.lean b/Mathlib/Data/List/DropRight.lean index 1d2f505245b4ce..4a752eebdb95a5 100644 --- a/Mathlib/Data/List/DropRight.lean +++ b/Mathlib/Data/List/DropRight.lean @@ -10,7 +10,6 @@ public import Mathlib.Data.List.TakeWhile public import Mathlib.Data.List.Basic /-! - # Dropping or taking from lists on the right Taking or removing element from the tail end of a list diff --git a/Mathlib/Data/List/ToFinsupp.lean b/Mathlib/Data/List/ToFinsupp.lean index d7499e9849650d..79b60d4666b89f 100644 --- a/Mathlib/Data/List/ToFinsupp.lean +++ b/Mathlib/Data/List/ToFinsupp.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Group.Nat.Defs public import Mathlib.Data.List.GetD /-! - # Lists as finsupp ## Main definitions diff --git a/Mathlib/Data/Nat/Choose/Vandermonde.lean b/Mathlib/Data/Nat/Choose/Vandermonde.lean index 771eb08dd0cd10..30a50e0bbdd37f 100644 --- a/Mathlib/Data/Nat/Choose/Vandermonde.lean +++ b/Mathlib/Data/Nat/Choose/Vandermonde.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Polynomial.Coeff public import Mathlib.Data.Nat.Choose.Basic /-! - # Vandermonde's identity In this file we prove Vandermonde's identity (`Nat.add_choose_eq`): diff --git a/Mathlib/Data/PEquiv.lean b/Mathlib/Data/PEquiv.lean index 8e7fa356c79030..1ce0f1b6ef086b 100644 --- a/Mathlib/Data/PEquiv.lean +++ b/Mathlib/Data/PEquiv.lean @@ -11,7 +11,6 @@ public import Mathlib.Data.Set.Basic public import Mathlib.Tactic.Contrapose /-! - # Partial Equivalences In this file, we define partial equivalences `PEquiv`, which are a bijection between a subset of `α` diff --git a/Mathlib/Data/QPF/Univariate/Basic.lean b/Mathlib/Data/QPF/Univariate/Basic.lean index 86ffd37c8f889c..ae6951499f1509 100644 --- a/Mathlib/Data/QPF/Univariate/Basic.lean +++ b/Mathlib/Data/QPF/Univariate/Basic.lean @@ -8,7 +8,6 @@ module public import Mathlib.Data.PFunctor.Univariate.M /-! - # Quotients of Polynomial Functors We assume the following: diff --git a/Mathlib/Data/Set/Basic.lean b/Mathlib/Data/Set/Basic.lean index f351d53127ac98..b6886ed5c19adb 100644 --- a/Mathlib/Data/Set/Basic.lean +++ b/Mathlib/Data/Set/Basic.lean @@ -490,7 +490,6 @@ theorem empty_ssubset : ∅ ⊂ s ↔ s.Nonempty := alias ⟨_, Nonempty.empty_ssubset⟩ := empty_ssubset /-! - ### Universal set. In Lean `@univ α` (or `univ : Set α`) is the set that contains all elements of type `α`. diff --git a/Mathlib/Data/Set/Subset.lean b/Mathlib/Data/Set/Subset.lean index ec097dbab8d9b6..f55a4a85bb39f7 100644 --- a/Mathlib/Data/Set/Subset.lean +++ b/Mathlib/Data/Set/Subset.lean @@ -29,7 +29,6 @@ This notation, (together with the `↑` notation for `Set.CoeHead`) is defined in `Mathlib/Data/Set/Notation.lean` and is scoped to the `Set.Notation` namespace. To enable it, use `open Set.Notation`. - ## Naming conventions Theorem names refer to `↓∩` as `preimage_val`. diff --git a/Mathlib/Data/Sym/Card.lean b/Mathlib/Data/Sym/Card.lean index 6858fdc88e0403..5ae2b7a1194623 100644 --- a/Mathlib/Data/Sym/Card.lean +++ b/Mathlib/Data/Sym/Card.lean @@ -47,7 +47,6 @@ multisets of size `k` over `α`. Thus the statement that these are counted by `m while the "stars and bars" technique gives `Sym.card_sym_eq_choose : card (Sym α k) = choose (card α + k - 1) k` - ## Tags stars and bars, multichoose diff --git a/Mathlib/Data/TypeVec.lean b/Mathlib/Data/TypeVec.lean index faf2abfe4dfdc1..bbb9dd8d2f30f2 100644 --- a/Mathlib/Data/TypeVec.lean +++ b/Mathlib/Data/TypeVec.lean @@ -10,7 +10,6 @@ public import Mathlib.Logic.Function.Basic public import Mathlib.Tactic.Common /-! - # Tuples of types, and their categorical structure. ## Features diff --git a/Mathlib/Data/ZMod/Basic.lean b/Mathlib/Data/ZMod/Basic.lean index 492558652011b0..9f3e66844b3249 100644 --- a/Mathlib/Data/ZMod/Basic.lean +++ b/Mathlib/Data/ZMod/Basic.lean @@ -17,7 +17,6 @@ public import Mathlib.Tactic.FinCases Definition of the integers mod n, and the field structure on the integers mod p. - ## Definitions * `ZMod n`, which is for integers modulo a nat `n : ℕ` diff --git a/Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean b/Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean index 5ab181c92b7d3c..6d47684d55ade5 100644 --- a/Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean +++ b/Mathlib/Dynamics/TopologicalEntropy/Semiconj.lean @@ -12,7 +12,6 @@ public import Mathlib.Dynamics.TopologicalEntropy.CoverEntropy Consider two dynamical systems `(X, S)` and `(Y, T)` together with a semiconjugacy `φ`: - ``` X ---S--> X | | diff --git a/Mathlib/FieldTheory/Finite/Extension.lean b/Mathlib/FieldTheory/Finite/Extension.lean index 299773369de76c..5441b2cffba2c6 100644 --- a/Mathlib/FieldTheory/Finite/Extension.lean +++ b/Mathlib/FieldTheory/Finite/Extension.lean @@ -18,7 +18,6 @@ non-unique isomorphism) extension `l` of `k` of any given degree `n > 0`. This extension is Galois with cyclic Galois group of degree `n`, and the (arithmetic) Frobenius map `x ↦ x ^ q` is a generator. - ## Main definition * `FiniteField.Extension k p n` is a non-canonically chosen extension of `k` of degree `n` diff --git a/Mathlib/FieldTheory/Galois/Abelian.lean b/Mathlib/FieldTheory/Galois/Abelian.lean index 640310a1c5c8b4..6c20f9d2efb025 100644 --- a/Mathlib/FieldTheory/Galois/Abelian.lean +++ b/Mathlib/FieldTheory/Galois/Abelian.lean @@ -8,7 +8,6 @@ module public import Mathlib.FieldTheory.Galois.Infinite /-! - # Abelian extensions In this file, we define the typeclass of abelian extensions and provide some basic API. diff --git a/Mathlib/FieldTheory/Galois/GaloisClosure.lean b/Mathlib/FieldTheory/Galois/GaloisClosure.lean index dfc8954599d3c3..3da0d5af269a6e 100644 --- a/Mathlib/FieldTheory/Galois/GaloisClosure.lean +++ b/Mathlib/FieldTheory/Galois/GaloisClosure.lean @@ -8,7 +8,6 @@ module public import Mathlib.FieldTheory.Galois.Basic /-! - # Main definitions and results In a field extension `K/k` diff --git a/Mathlib/FieldTheory/Galois/Infinite.lean b/Mathlib/FieldTheory/Galois/Infinite.lean index ad9b29753d6c4d..f819684647f8fb 100644 --- a/Mathlib/FieldTheory/Galois/Infinite.lean +++ b/Mathlib/FieldTheory/Galois/Infinite.lean @@ -9,7 +9,6 @@ public import Mathlib.FieldTheory.KrullTopology public import Mathlib.FieldTheory.Galois.GaloisClosure public import Mathlib.Topology.Algebra.Group.ClosedSubgroup /-! - # The Fundamental Theorem of Infinite Galois Theory In this file, we prove the fundamental theorem of infinite Galois theory and the special case for diff --git a/Mathlib/FieldTheory/Galois/Profinite.lean b/Mathlib/FieldTheory/Galois/Profinite.lean index 8c3ec86b00cdbf..e47f00ed34550a 100644 --- a/Mathlib/FieldTheory/Galois/Profinite.lean +++ b/Mathlib/FieldTheory/Galois/Profinite.lean @@ -10,7 +10,6 @@ public import Mathlib.FieldTheory.Galois.GaloisClosure public import Mathlib.Topology.Algebra.Category.ProfiniteGrp.Basic /-! - # Galois Group as a profinite group In this file, we prove that given a field extension `K/k`, there is a continuous isomorphism between diff --git a/Mathlib/FieldTheory/IsPerfectClosure.lean b/Mathlib/FieldTheory/IsPerfectClosure.lean index 2038d7e583b0e3..9122594440261f 100644 --- a/Mathlib/FieldTheory/IsPerfectClosure.lean +++ b/Mathlib/FieldTheory/IsPerfectClosure.lean @@ -9,7 +9,6 @@ public import Mathlib.FieldTheory.PurelyInseparable.Basic public import Mathlib.FieldTheory.PerfectClosure /-! - # `IsPerfectClosure` predicate This file contains `IsPerfectClosure` which asserts that `L` is a perfect closure of `K` under a diff --git a/Mathlib/FieldTheory/LinearDisjoint.lean b/Mathlib/FieldTheory/LinearDisjoint.lean index 5894be8689642f..d10f8aec44cf46 100644 --- a/Mathlib/FieldTheory/LinearDisjoint.lean +++ b/Mathlib/FieldTheory/LinearDisjoint.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.AlgebraicIndependent.RankAndCardinality public import Mathlib.RingTheory.LinearDisjoint /-! - # Linearly disjoint fields This file contains basics about the linearly disjoint fields. diff --git a/Mathlib/FieldTheory/NormalizedTrace.lean b/Mathlib/FieldTheory/NormalizedTrace.lean index 9947816d395b90..32cf821ae36425 100644 --- a/Mathlib/FieldTheory/NormalizedTrace.lean +++ b/Mathlib/FieldTheory/NormalizedTrace.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.Trace.Basic /-! - # Normalized trace This file defines the *normalized trace* map; that is, an `F`-linear map from the algebraic closure diff --git a/Mathlib/FieldTheory/Perfect.lean b/Mathlib/FieldTheory/Perfect.lean index b3987a9aa0062b..08a2877f05e7a2 100644 --- a/Mathlib/FieldTheory/Perfect.lean +++ b/Mathlib/FieldTheory/Perfect.lean @@ -11,7 +11,6 @@ public import Mathlib.FieldTheory.KummerPolynomial public import Mathlib.FieldTheory.Separable /-! - # Perfect fields and rings In this file we define perfect fields, together with a generalisation to (commutative) rings in diff --git a/Mathlib/FieldTheory/PerfectClosure.lean b/Mathlib/FieldTheory/PerfectClosure.lean index a8d1112e9e7599..3bc615766e02ef 100644 --- a/Mathlib/FieldTheory/PerfectClosure.lean +++ b/Mathlib/FieldTheory/PerfectClosure.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.CharP.Lemmas public import Mathlib.FieldTheory.Perfect /-! - # The perfect closure of a characteristic `p` ring ## Main definitions diff --git a/Mathlib/FieldTheory/PurelyInseparable/Basic.lean b/Mathlib/FieldTheory/PurelyInseparable/Basic.lean index e737a97e815e7d..bc1d47afc94876 100644 --- a/Mathlib/FieldTheory/PurelyInseparable/Basic.lean +++ b/Mathlib/FieldTheory/PurelyInseparable/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.CharP.IntermediateField public import Mathlib.FieldTheory.IsSepClosed /-! - # Basic results about purely inseparable extensions This file contains basic definitions and results about purely inseparable extensions. diff --git a/Mathlib/FieldTheory/PurelyInseparable/Exponent.lean b/Mathlib/FieldTheory/PurelyInseparable/Exponent.lean index 3243f8425492a4..2645bb3f046a9f 100644 --- a/Mathlib/FieldTheory/PurelyInseparable/Exponent.lean +++ b/Mathlib/FieldTheory/PurelyInseparable/Exponent.lean @@ -8,7 +8,6 @@ module public import Mathlib.FieldTheory.PurelyInseparable.Basic /-! - # The exponent of purely inseparable extensions This file defines the exponent of a purely inseparable extension (if one exists) and diff --git a/Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean b/Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean index d22f0846dcb214..ed5397ed4853c2 100644 --- a/Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean +++ b/Mathlib/FieldTheory/PurelyInseparable/PerfectClosure.lean @@ -11,7 +11,6 @@ public import Mathlib.FieldTheory.PurelyInseparable.Basic public import Mathlib.LinearAlgebra.Dimension.OrzechProperty /-! - # Basic results about relative perfect closure This file contains basic results about relative perfect closures. diff --git a/Mathlib/FieldTheory/PurelyInseparable/Tower.lean b/Mathlib/FieldTheory/PurelyInseparable/Tower.lean index 832583fd13a49b..3aa94127ec4ae3 100644 --- a/Mathlib/FieldTheory/PurelyInseparable/Tower.lean +++ b/Mathlib/FieldTheory/PurelyInseparable/Tower.lean @@ -9,7 +9,6 @@ public import Mathlib.FieldTheory.LinearDisjoint public import Mathlib.FieldTheory.PurelyInseparable.PerfectClosure /-! - # Tower law for purely inseparable extensions This file contains results related to `Field.sepDegree`, `Field.insepDegree` and the tower law. diff --git a/Mathlib/FieldTheory/Relrank.lean b/Mathlib/FieldTheory/Relrank.lean index 0c5de1459a4ff3..6af96c24e0d82f 100644 --- a/Mathlib/FieldTheory/Relrank.lean +++ b/Mathlib/FieldTheory/Relrank.lean @@ -8,7 +8,6 @@ module public import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic /-! - # Relative rank of subfields and intermediate fields This file contains basics about the relative rank of subfields and intermediate fields. diff --git a/Mathlib/FieldTheory/Separable.lean b/Mathlib/FieldTheory/Separable.lean index 6fe3e6458fa074..6af7ee857e3a86 100644 --- a/Mathlib/FieldTheory/Separable.lean +++ b/Mathlib/FieldTheory/Separable.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.PowerBasis public import Mathlib.Data.ENat.Lattice /-! - # Separable polynomials We define a polynomial to be separable if it is coprime with its derivative. We prove basic diff --git a/Mathlib/FieldTheory/SeparableClosure.lean b/Mathlib/FieldTheory/SeparableClosure.lean index 43518d5675b070..d5cd1915f057f8 100644 --- a/Mathlib/FieldTheory/SeparableClosure.lean +++ b/Mathlib/FieldTheory/SeparableClosure.lean @@ -9,7 +9,6 @@ public import Mathlib.FieldTheory.SeparableDegree public import Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure /-! - # Separable closure This file contains basics about the (relative) separable closure of a field extension. diff --git a/Mathlib/FieldTheory/SeparableDegree.lean b/Mathlib/FieldTheory/SeparableDegree.lean index 690441f0b98017..1a5eabde8306b4 100644 --- a/Mathlib/FieldTheory/SeparableDegree.lean +++ b/Mathlib/FieldTheory/SeparableDegree.lean @@ -16,7 +16,6 @@ public import Mathlib.RingTheory.Polynomial.UniqueFactorization /-! - # Separable degree This file contains basics about the separable degree of a field extension. diff --git a/Mathlib/FieldTheory/SeparablyGenerated.lean b/Mathlib/FieldTheory/SeparablyGenerated.lean index 862f674dbacdd6..f84c22fad0dcc5 100644 --- a/Mathlib/FieldTheory/SeparablyGenerated.lean +++ b/Mathlib/FieldTheory/SeparablyGenerated.lean @@ -14,7 +14,6 @@ public import Mathlib.RingTheory.AlgebraicIndependent.AlgebraicClosure public import Mathlib.RingTheory.Polynomial.GaussLemma /-! - # Separably generated extensions We aim to formalize the following result: diff --git a/Mathlib/Geometry/Convex/Cone/Pointed.lean b/Mathlib/Geometry/Convex/Cone/Pointed.lean index f2370255b2ba01..531596ce69057f 100644 --- a/Mathlib/Geometry/Convex/Cone/Pointed.lean +++ b/Mathlib/Geometry/Convex/Cone/Pointed.lean @@ -218,7 +218,6 @@ variable [AddCommMonoid F] [Module R F] variable [AddCommMonoid G] [Module R G] /-! - ## Maps between pointed cones There is already a definition of maps between submodules, `Submodule.map`. In our case, these maps diff --git a/Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean b/Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean index 97ef839a3bc66e..1d3e173b30040f 100644 --- a/Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean +++ b/Mathlib/Geometry/Manifold/Algebra/LeftInvariantDerivation.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Derivation.Lie public import Mathlib.Geometry.Manifold.DerivationBundle /-! - # Left invariant derivations In this file we define the concept of left invariant derivations for a Lie group. The concept is diff --git a/Mathlib/Geometry/Manifold/Algebra/LieGroup.lean b/Mathlib/Geometry/Manifold/Algebra/LieGroup.lean index 74cf7163b9907f..ce4b419f707cd9 100644 --- a/Mathlib/Geometry/Manifold/Algebra/LieGroup.lean +++ b/Mathlib/Geometry/Manifold/Algebra/LieGroup.lean @@ -26,7 +26,6 @@ that Lie groups here are not necessarily finite dimensional. * `ContMDiffInv₀`: typeclass for `C^n` manifolds with `0` and `Inv` such that inversion is `C^n` map at each non-zero point. This includes complete normed fields and (multiplicative) Lie groups. - ## Main results * `ContMDiff.inv`, `ContMDiff.div` and variants: point-wise inversion and division of maps `M → G` is `C^n`. diff --git a/Mathlib/Geometry/Manifold/DerivationBundle.lean b/Mathlib/Geometry/Manifold/DerivationBundle.lean index 7c2cdf3003dd13..00ce3bd406128a 100644 --- a/Mathlib/Geometry/Manifold/DerivationBundle.lean +++ b/Mathlib/Geometry/Manifold/DerivationBundle.lean @@ -9,7 +9,6 @@ public import Mathlib.Geometry.Manifold.Algebra.SmoothFunctions public import Mathlib.RingTheory.Derivation.Basic /-! - # Derivation bundle In this file we define the derivations at a point of a manifold on the algebra of smooth functions. diff --git a/Mathlib/Geometry/Manifold/Instances/Sphere.lean b/Mathlib/Geometry/Manifold/Instances/Sphere.lean index bd2d69431eba6d..b6a02e26b325a7 100644 --- a/Mathlib/Geometry/Manifold/Instances/Sphere.lean +++ b/Mathlib/Geometry/Manifold/Instances/Sphere.lean @@ -50,7 +50,6 @@ centred at `0` of radius `1`) with the following structure: We furthermore show that `Circle.exp` (defined in `Analysis.Complex.Circle` to be the natural map `fun t ↦ exp (t * I)` from `ℝ` to `Circle`) is analytic. - ## Implementation notes The model space for the charted space instance is `EuclideanSpace ℝ (Fin n)`, where `n` is a diff --git a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean index 4175f420b82bb0..b92d600f134587 100644 --- a/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean +++ b/Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean @@ -9,7 +9,6 @@ public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace public import Mathlib.RingTheory.LocalRing.ResidueField.Basic /-! - # Residue fields of points Any point `x` of a locally ringed space `X` comes with a natural residue field, namely the residue diff --git a/Mathlib/GroupTheory/FreeGroup/Basic.lean b/Mathlib/GroupTheory/FreeGroup/Basic.lean index 614f4742809c6c..530e74366e9e33 100644 --- a/Mathlib/GroupTheory/FreeGroup/Basic.lean +++ b/Mathlib/GroupTheory/FreeGroup/Basic.lean @@ -47,7 +47,6 @@ over `FreeGroup.Red.Step`. For the additive version we introduce the same relation under a different name so that we can distinguish the quotient types more easily. - ## Tags free group, Newman's diamond lemma, Church-Rosser theorem diff --git a/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean b/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean index d91e0dc0e1c45b..787b31290549d2 100644 --- a/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean +++ b/Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Group.Subgroup.Lattice public import Mathlib.GroupTheory.GroupAction.FixedPoints /-! - # Fixing submonoid, fixing subgroup of an action In the presence of an action of a monoid or a group, diff --git a/Mathlib/GroupTheory/GroupAction/MultipleTransitivity.lean b/Mathlib/GroupTheory/GroupAction/MultipleTransitivity.lean index 102af46eb06390..e61f4547edb995 100644 --- a/Mathlib/GroupTheory/GroupAction/MultipleTransitivity.lean +++ b/Mathlib/GroupTheory/GroupAction/MultipleTransitivity.lean @@ -42,7 +42,6 @@ public import Mathlib.SetTheory.Cardinal.Arithmetic * `SubMulAction.ofStabilizer.isMultiplyPretransitive_iff_conj hg` shows the same result for `n`-transitivity. - * `SubMulAction.ofStabilizer.isMultiplyPretransitive_iff` : if the action of `G` on `α` is pretransitive, then it is `n.succ` pretransitive if and only if the action of `stabilizer G a` on `ofStabilizer G a` is `n`-pretransitive. diff --git a/Mathlib/GroupTheory/GroupAction/SubMulAction.lean b/Mathlib/GroupTheory/GroupAction/SubMulAction.lean index 089f7bd78458f5..e1b63da68b1ad4 100644 --- a/Mathlib/GroupTheory/GroupAction/SubMulAction.lean +++ b/Mathlib/GroupTheory/GroupAction/SubMulAction.lean @@ -13,7 +13,6 @@ public import Mathlib.GroupTheory.GroupAction.Defs public import Mathlib.GroupTheory.GroupAction.Hom /-! - # Sets invariant to a `MulAction` In this file we define `SubMulAction R M`; a subset of a `MulAction R M` which is closed with diff --git a/Mathlib/GroupTheory/HNNExtension.lean b/Mathlib/GroupTheory/HNNExtension.lean index 181ccc15c4e004..5eb90f2ee1ef4b 100644 --- a/Mathlib/GroupTheory/HNNExtension.lean +++ b/Mathlib/GroupTheory/HNNExtension.lean @@ -11,7 +11,6 @@ public import Mathlib.GroupTheory.Coprod.Basic public import Mathlib.GroupTheory.Complement /-! - ## HNN Extensions of Groups This file defines the HNN extension of a group `G`, `HNNExtension G A B φ`. Given a group `G`, diff --git a/Mathlib/GroupTheory/MonoidLocalization/Cardinality.lean b/Mathlib/GroupTheory/MonoidLocalization/Cardinality.lean index bd0571070236ec..e079eb3aa37875 100644 --- a/Mathlib/GroupTheory/MonoidLocalization/Cardinality.lean +++ b/Mathlib/GroupTheory/MonoidLocalization/Cardinality.lean @@ -9,7 +9,6 @@ public import Mathlib.GroupTheory.MonoidLocalization.Basic public import Mathlib.GroupTheory.OreLocalization.Cardinality /-! - # Cardinality of localizations of commutative monoids This file contains some results on cardinality of localizations. diff --git a/Mathlib/GroupTheory/Nilpotent.lean b/Mathlib/GroupTheory/Nilpotent.lean index f8c36afe217b70..8250465280c225 100644 --- a/Mathlib/GroupTheory/Nilpotent.lean +++ b/Mathlib/GroupTheory/Nilpotent.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Group.Subgroup.Order public import Mathlib.GroupTheory.Commutator.Finite /-! - # Nilpotent groups An API for nilpotent groups, that is, groups for which the upper central series @@ -64,7 +63,6 @@ subgroup `G` of `G`, and `⊥` denotes the trivial subgroup `{1}`. is derived from that. * `IsNilpotent.to_isSolvable`: If `G` is nilpotent, it is solvable. - ## Warning A "central series" is usually defined to be a finite sequence of normal subgroups going diff --git a/Mathlib/GroupTheory/OreLocalization/Basic.lean b/Mathlib/GroupTheory/OreLocalization/Basic.lean index 41d9c575b2b182..ac3905cb7fae12 100644 --- a/Mathlib/GroupTheory/OreLocalization/Basic.lean +++ b/Mathlib/GroupTheory/OreLocalization/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.Algebra.Group.Units.Defs public import Mathlib.Algebra.Group.Basic /-! - # Localization over left Ore sets. This file defines the localization of a monoid over a left Ore set and proves its universal @@ -29,7 +28,6 @@ a denominator `s : S`. * * [Zoran Škoda, *Noncommutative localization in noncommutative geometry*][skoda2006] - ## Tags localization, Ore, non-commutative diff --git a/Mathlib/GroupTheory/OreLocalization/Cardinality.lean b/Mathlib/GroupTheory/OreLocalization/Cardinality.lean index 92d6feb5aa44ce..ec1d755ee5df71 100644 --- a/Mathlib/GroupTheory/OreLocalization/Cardinality.lean +++ b/Mathlib/GroupTheory/OreLocalization/Cardinality.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.OreLocalization.Basic public import Mathlib.SetTheory.Cardinal.Arithmetic /-! - # Cardinality of Ore localizations This file contains some results on cardinality of Ore localizations. diff --git a/Mathlib/GroupTheory/OreLocalization/OreSet.lean b/Mathlib/GroupTheory/OreLocalization/OreSet.lean index 2829f98073cdbc..b6b0530ca2e710 100644 --- a/Mathlib/GroupTheory/OreLocalization/OreSet.lean +++ b/Mathlib/GroupTheory/OreLocalization/OreSet.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Group.Submonoid.Defs /-! - # (Left) Ore sets This defines left Ore sets on arbitrary monoids. diff --git a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean index 8cb2d346aea3a7..da85398efed1f6 100644 --- a/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean +++ b/Mathlib/GroupTheory/Perm/Cycle/Concrete.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.Perm.Cycle.Type public import Mathlib.GroupTheory.Perm.List /-! - # Properties of cyclic permutations constructed from lists/cycles In the following, `{α : Type*} [Fintype α] [DecidableEq α]`. diff --git a/Mathlib/GroupTheory/PushoutI.lean b/Mathlib/GroupTheory/PushoutI.lean index 64677d3c0e3f85..6661809e4a98c4 100644 --- a/Mathlib/GroupTheory/PushoutI.lean +++ b/Mathlib/GroupTheory/PushoutI.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.Coprod.Basic public import Mathlib.GroupTheory.Complement /-! - ## Pushouts of Monoids and Groups This file defines wide pushouts of monoids and groups and proves some properties diff --git a/Mathlib/GroupTheory/SpecificGroups/Alternating.lean b/Mathlib/GroupTheory/SpecificGroups/Alternating.lean index 13c0a8012a4303..1e32960e749f7c 100644 --- a/Mathlib/GroupTheory/SpecificGroups/Alternating.lean +++ b/Mathlib/GroupTheory/SpecificGroups/Alternating.lean @@ -52,7 +52,6 @@ consisting of the even permutations. ## Tags alternating group permutation simple characteristic index - ## TODO * Show that `alternatingGroup α` is simple if and only if `Fintype.card α ≠ 4`. diff --git a/Mathlib/GroupTheory/Submonoid/Inverses.lean b/Mathlib/GroupTheory/Submonoid/Inverses.lean index d4efa62017f0eb..71cbed03d4f302 100644 --- a/Mathlib/GroupTheory/Submonoid/Inverses.lean +++ b/Mathlib/GroupTheory/Submonoid/Inverses.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Group.Submonoid.Pointwise /-! - # Submonoid of inverses Given a submonoid `N` of a monoid `M`, we define the submonoid `N.leftInv` as the submonoid of diff --git a/Mathlib/Lean/Meta/KAbstractPositions.lean b/Mathlib/Lean/Meta/KAbstractPositions.lean index 55291bb6c9111e..1eb4c52c7fcff1 100644 --- a/Mathlib/Lean/Meta/KAbstractPositions.lean +++ b/Mathlib/Lean/Meta/KAbstractPositions.lean @@ -11,7 +11,6 @@ public import Lean.Meta.ExprLens public import Lean.Meta.Check /-! - # Find the positions of a pattern in an expression This file defines some tools for dealing with subexpressions and occurrence numbers. diff --git a/Mathlib/Lean/Meta/RefinedDiscrTree.lean b/Mathlib/Lean/Meta/RefinedDiscrTree.lean index e51f6df86354dd..13a30216bca7fb 100644 --- a/Mathlib/Lean/Meta/RefinedDiscrTree.lean +++ b/Mathlib/Lean/Meta/RefinedDiscrTree.lean @@ -58,7 +58,6 @@ and includes many more features. Depending on the configuration, β-reduction and ζ-reduction may be disabled, so the resulting applied lambda expressions or let-expressions are indexed by `Key.opaque`. - ## Lazy computation To encode an `Expr` as a sequence of `Key`s, we start with a `LazyEntry` and @@ -69,7 +68,6 @@ and lazy entries, or returns `none` if the last key has been reached already. The `RefinedDiscrTree` then stores these `LazyEntries` at its leafs, and evaluates them only if the lookup algorithm reaches this leaf. - ## Alternative optimizations `RefinedDiscrTree` is a non-persistent lazy data-structure. Therefore, when using it, you should @@ -85,7 +83,6 @@ different lemmas anyways. Make a version of `RefinedDiscrTree` that is optimal for tactics like `simp` and `fun_prop`. This would mean using a persistent data structure, and possibly a non-lazy structure. - ## Matching vs Unification A discrimination tree can be used in two ways: either with (unification) or without (matching) diff --git a/Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean b/Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean index 628935e785ca77..64d4eb7250fbee 100644 --- a/Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean +++ b/Mathlib/LinearAlgebra/BilinearForm/DualLattice.lean @@ -8,7 +8,6 @@ module public import Mathlib.LinearAlgebra.BilinearForm.Properties /-! - # Dual submodule with respect to a bilinear form. ## Main definitions and results diff --git a/Mathlib/LinearAlgebra/Charpoly/Basic.lean b/Mathlib/LinearAlgebra/Charpoly/Basic.lean index c3e7afb498ccc3..8b0b8d9e81cee7 100644 --- a/Mathlib/LinearAlgebra/Charpoly/Basic.lean +++ b/Mathlib/LinearAlgebra/Charpoly/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.FieldTheory.Minpoly.Field public import Mathlib.LinearAlgebra.Determinant /-! - # Characteristic polynomial We define the characteristic polynomial of `f : M →ₗ[R] M`, where `M` is a finite and diff --git a/Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean b/Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean index a46a5585149985..c623e240c9cc83 100644 --- a/Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean +++ b/Mathlib/LinearAlgebra/Charpoly/ToMatrix.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Matrix.Basis public import Mathlib.RingTheory.Finiteness.Prod /-! - # Characteristic polynomial ## Main result diff --git a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean index 8907719c5773e6..d2bc8e6248ede6 100644 --- a/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean +++ b/Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean @@ -9,7 +9,6 @@ public import Mathlib.LinearAlgebra.Dimension.Constructions public import Mathlib.LinearAlgebra.Dimension.Subsingleton /-! - # Some results on free modules over rings satisfying strong rank condition This file contains some results on free modules over rings satisfying strong rank condition. diff --git a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean index 1d820f2b0b100c..3df0b49e9549d2 100644 --- a/Mathlib/LinearAlgebra/Dimension/RankNullity.lean +++ b/Mathlib/LinearAlgebra/Dimension/RankNullity.lean @@ -11,7 +11,6 @@ public import Mathlib.LinearAlgebra.Isomorphisms public import Mathlib.Logic.Equiv.Fin.Rotate /-! - # The rank nullity theorem In this file we provide the rank nullity theorem as a typeclass, and prove various corollaries diff --git a/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean b/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean index 37c5aa544d2144..9589a8d906a827 100644 --- a/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean +++ b/Mathlib/LinearAlgebra/Eigenspace/Matrix.lean @@ -8,7 +8,6 @@ module public import Mathlib.LinearAlgebra.Eigenspace.Basic /-! - # Eigenvalues, Eigenvectors and Spectrum for Matrices This file collects results about eigenvectors, eigenvalues and spectrum specific to matrices diff --git a/Mathlib/LinearAlgebra/FiniteSpan.lean b/Mathlib/LinearAlgebra/FiniteSpan.lean index 97be471f0fb99c..b8d51da0f14eaf 100644 --- a/Mathlib/LinearAlgebra/FiniteSpan.lean +++ b/Mathlib/LinearAlgebra/FiniteSpan.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Span.Defs public import Mathlib.Algebra.Module.Equiv.Basic /-! - # Additional results about finite spanning sets in linear algebra -/ diff --git a/Mathlib/LinearAlgebra/FreeModule/StrongRankCondition.lean b/Mathlib/LinearAlgebra/FreeModule/StrongRankCondition.lean index 6d6e235b901027..6d303179472259 100644 --- a/Mathlib/LinearAlgebra/FreeModule/StrongRankCondition.lean +++ b/Mathlib/LinearAlgebra/FreeModule/StrongRankCondition.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.FiniteType public import Mathlib.LinearAlgebra.InvariantBasisNumber /-! - # Strong rank condition for commutative rings We provide a shortcut instance for the fact that any nontrivial commutative ring satisfies diff --git a/Mathlib/LinearAlgebra/InvariantBasisNumber.lean b/Mathlib/LinearAlgebra/InvariantBasisNumber.lean index 61512f8a35ca17..396ebd0c79f592 100644 --- a/Mathlib/LinearAlgebra/InvariantBasisNumber.lean +++ b/Mathlib/LinearAlgebra/InvariantBasisNumber.lean @@ -37,7 +37,6 @@ It is also useful to consider the following stronger conditions: states that for any finitely generated `R`-module `M`, any surjective homomorphism `f : N → M` from a submodule `N` of `M` to `M` is injective. - ## Instances - `IsNoetherianRing.orzechProperty` (defined in `Mathlib/RingTheory/Noetherian/Orzech.lean`) : @@ -62,7 +61,6 @@ More generally, every commutative ring satisfies the Orzech property, hence the strong rank condition, which is proved in `Mathlib/RingTheory/FiniteType.lean`. We keep `invariantBasisNumber_of_nontrivial_of_commRing` here since it imports fewer files. - ## Counterexamples to converse results The following examples can be found in the book of Lam [lam_1999] @@ -74,7 +72,6 @@ The following examples can be found in the book of Lam [lam_1999] two-sided ideal `(ac − 1, bd − 1, ab, cd)` satisfies the invariant basis number property but not the rank condition. - ## Future work So far, there is no API at all for the `InvariantBasisNumber` class. There are several natural diff --git a/Mathlib/LinearAlgebra/LeftExact.lean b/Mathlib/LinearAlgebra/LeftExact.lean index 86066438f79800..a07bcf63282a04 100644 --- a/Mathlib/LinearAlgebra/LeftExact.lean +++ b/Mathlib/LinearAlgebra/LeftExact.lean @@ -11,14 +11,12 @@ public import Mathlib.LinearAlgebra.BilinearMap /-! # The Left Exactness of Hom - If `M1 → M2 → M3 → 0` is an exact sequence of `R`-modules and `N` is an `R`-module, then `0 → (M3 →ₗ[R] N) → (M2 →ₗ[R] N) → (M1 →ₗ[R] N)` is exact. In this file, we show the exactness at `M2 →ₗ[R] N` (`exact_lcomp_of_exact_of_surjective`); the injectivity part is `LinearMap.lcomp_injective_of_surjective` in the file `Mathlib.LinearAlgebra.BilinearMap`. - -/ public section diff --git a/Mathlib/LinearAlgebra/LinearDisjoint.lean b/Mathlib/LinearAlgebra/LinearDisjoint.lean index 92b016cadf3c42..a035cc35cecd58 100644 --- a/Mathlib/LinearAlgebra/LinearDisjoint.lean +++ b/Mathlib/LinearAlgebra/LinearDisjoint.lean @@ -12,7 +12,6 @@ public import Mathlib.LinearAlgebra.Dimension.Finite public import Mathlib.RingTheory.Flat.Basic /-! - # Linearly disjoint submodules This file contains basics about linearly disjoint submodules. diff --git a/Mathlib/LinearAlgebra/LinearIndependent/BaseChange.lean b/Mathlib/LinearAlgebra/LinearIndependent/BaseChange.lean index ad95ef5fa94600..cb6197d9353345 100644 --- a/Mathlib/LinearAlgebra/LinearIndependent/BaseChange.lean +++ b/Mathlib/LinearAlgebra/LinearIndependent/BaseChange.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.TensorProduct.Basis public import Mathlib.RingTheory.TensorProduct.IsBaseChangeFree /-! - # Base change for linear independence This file is a place to collect base change results for linear independence. diff --git a/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean b/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean index 057bfa60361098..aaedacab4f631c 100644 --- a/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean +++ b/Mathlib/LinearAlgebra/LinearIndependent/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Finsupp.LinearCombination public meta import Mathlib.Lean.Expr.ExtraRecognizers /-! - # Linear independence This file defines linear independence in a module or vector space. diff --git a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean index d9cb65f1a9e774..26bf0a7a6bb0a8 100644 --- a/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean +++ b/Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Matrix.ToLin public import Mathlib.Algebra.Module.SpanRank /-! - # Cayley-Hamilton theorem for f.g. modules. Given a fixed finite spanning set `b : ι → M` of an `R`-module `M`, we say that a matrix `M` diff --git a/Mathlib/LinearAlgebra/Projectivization/Basic.lean b/Mathlib/LinearAlgebra/Projectivization/Basic.lean index c691128f5015f6..19d73109ce0226 100644 --- a/Mathlib/LinearAlgebra/Projectivization/Basic.lean +++ b/Mathlib/LinearAlgebra/Projectivization/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition public import Mathlib.LinearAlgebra.FiniteDimensional.Basic /-! - # Projective Spaces This file contains the definition of the projectivization of a vector space over a field, diff --git a/Mathlib/LinearAlgebra/Projectivization/Constructions.lean b/Mathlib/LinearAlgebra/Projectivization/Constructions.lean index 04305fad079619..da344bfeae1290 100644 --- a/Mathlib/LinearAlgebra/Projectivization/Constructions.lean +++ b/Mathlib/LinearAlgebra/Projectivization/Constructions.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.Matrix.DotProduct public import Mathlib.LinearAlgebra.Projectivization.Basic /-! - # Dot Product and Cross Product on Projective Spaces This file defines the dot product and cross product on projective spaces. diff --git a/Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean b/Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean index 9aae6b59288331..43f9413007e05e 100644 --- a/Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean +++ b/Mathlib/LinearAlgebra/TensorProduct/Finiteness.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Finiteness.Basic public import Mathlib.LinearAlgebra.TensorProduct.Map /-! - # Some finiteness results of tensor product This file contains some finiteness results of tensor product. diff --git a/Mathlib/LinearAlgebra/TensorProduct/Pi.lean b/Mathlib/LinearAlgebra/TensorProduct/Pi.lean index 99efa2312c3080..04c5031a8d61f4 100644 --- a/Mathlib/LinearAlgebra/TensorProduct/Pi.lean +++ b/Mathlib/LinearAlgebra/TensorProduct/Pi.lean @@ -9,7 +9,6 @@ public import Mathlib.LinearAlgebra.TensorProduct.Tower public import Mathlib.LinearAlgebra.Pi /-! - # Tensor product and products In this file we examine the behaviour of the tensor product with arbitrary and finite products. diff --git a/Mathlib/LinearAlgebra/TensorProduct/Quotient.lean b/Mathlib/LinearAlgebra/TensorProduct/Quotient.lean index d0595a460fb70d..a589451be0f26d 100644 --- a/Mathlib/LinearAlgebra/TensorProduct/Quotient.lean +++ b/Mathlib/LinearAlgebra/TensorProduct/Quotient.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Ideal.Maps public import Mathlib.RingTheory.Ideal.Quotient.Defs /-! - # Interaction between Quotients and Tensor Products This file contains constructions that relate quotients and tensor products. This file is also a home diff --git a/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean b/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean index 174859ad12e7f1..5fa479c81ee263 100644 --- a/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean +++ b/Mathlib/LinearAlgebra/TensorProduct/Subalgebra.lean @@ -9,7 +9,6 @@ public import Mathlib.LinearAlgebra.TensorProduct.Submodule public import Mathlib.RingTheory.TensorProduct.Maps /-! - # Some results on tensor product of subalgebras ## Linear maps induced by multiplication for subalgebras diff --git a/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean b/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean index cb790fb68c5164..91bc639233fb70 100644 --- a/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean +++ b/Mathlib/LinearAlgebra/TensorProduct/Submodule.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Algebra.Subalgebra.Lattice public import Mathlib.LinearAlgebra.DirectSum.Finsupp /-! - # Some results on tensor product of submodules ## Linear maps induced by multiplication for submodules diff --git a/Mathlib/Logic/Equiv/Option.lean b/Mathlib/Logic/Equiv/Option.lean index 6419012cdb2fb3..a100a53b441530 100644 --- a/Mathlib/Logic/Equiv/Option.lean +++ b/Mathlib/Logic/Equiv/Option.lean @@ -13,7 +13,6 @@ public import Mathlib.Logic.Equiv.Defs /-! # Equivalences for `Option α` - We define * `Equiv.optionCongr`: the `Option α ≃ Option β` constructed from `e : α ≃ β` by sending `none` to `none`, and applying `e` elsewhere. diff --git a/Mathlib/Logic/Function/FiberPartition.lean b/Mathlib/Logic/Function/FiberPartition.lean index f85ea3cb6bcf99..ceb0a5a582d99d 100644 --- a/Mathlib/Logic/Function/FiberPartition.lean +++ b/Mathlib/Logic/Function/FiberPartition.lean @@ -7,7 +7,6 @@ module public import Mathlib.Data.Set.Insert /-! - This file defines the type `f.Fiber` of fibers of a function `f : Y → Z`, and provides some API to work with and construct terms of this type. diff --git a/Mathlib/MeasureTheory/Function/AEEqFun.lean b/Mathlib/MeasureTheory/Function/AEEqFun.lean index 8a1138969e5bda..7572004f530b58 100644 --- a/Mathlib/MeasureTheory/Function/AEEqFun.lean +++ b/Mathlib/MeasureTheory/Function/AEEqFun.lean @@ -12,7 +12,6 @@ public import Mathlib.Order.Filter.Germ.Basic public import Mathlib.Topology.ContinuousMap.Algebra /-! - # Almost everywhere equal functions We build a space of equivalence classes of functions, where two functions are treated as identical @@ -63,7 +62,6 @@ See `Mathlib/MeasureTheory/Function/L1Space/AEEqFun.lean` for `L¹` space. * `comp₂`: Use `comp₂ g f₁ f₂` to get `[fun a ↦ g (f₁ a) (f₂ a)]`. For example, `[f + g]` is `comp₂ (+)` - ## Tags function space, almost everywhere equal, `L⁰`, ae_eq_fun diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean index 5a13702022a906..73e38eb7394fc8 100644 --- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean +++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean @@ -8,7 +8,6 @@ module public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic /-! - # Conditional expectation of indicator functions This file proves some results about the conditional expectation of an indicator function and diff --git a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean index 24c3958e3457f9..a42fb9ad86ad0b 100644 --- a/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean +++ b/Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean @@ -13,7 +13,6 @@ import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondJensen import Mathlib.MeasureTheory.Function.LpSeminorm.LpNorm /-! - # Conditional expectation of real-valued functions This file proves some results regarding the conditional expectation of real-valued functions. diff --git a/Mathlib/MeasureTheory/Measure/Portmanteau.lean b/Mathlib/MeasureTheory/Measure/Portmanteau.lean index e64bd9c2ba8bb3..9dfd9f1def7680 100644 --- a/Mathlib/MeasureTheory/Measure/Portmanteau.lean +++ b/Mathlib/MeasureTheory/Measure/Portmanteau.lean @@ -373,7 +373,6 @@ section LimitBorelImpliesLimsupClosedLE /-! ### Portmanteau implication: limit condition for Borel sets implies limsup for closed sets - In this section we prove, under the assumption that the underlying topological space `Ω` is pseudo-emetrizable, that @@ -482,7 +481,6 @@ section le_liminf_open_implies_convergence /-! ### Portmanteau implication: liminf condition for open sets implies weak convergence - In this section we prove for a sequence (μsₙ)ₙ Borel probability measures that (O) For any open set G, the liminf of the measures of G under μsₙ is at least diff --git a/Mathlib/MeasureTheory/Measure/Tight.lean b/Mathlib/MeasureTheory/Measure/Tight.lean index f701950d3f29f4..95b53bd60cc905 100644 --- a/Mathlib/MeasureTheory/Measure/Tight.lean +++ b/Mathlib/MeasureTheory/Measure/Tight.lean @@ -30,7 +30,6 @@ measures in the set, the complement of `K` has measure at most `ε`. * `isTightMeasureSet_singleton_of_innerRegularWRT`: every finite, inner-regular measure is tight. * `isTightMeasureSet_of_isCompact_closure`: every relatively compact set of measures is tight. - -/ @[expose] public section diff --git a/Mathlib/MeasureTheory/Measure/Trim.lean b/Mathlib/MeasureTheory/Measure/Trim.lean index 8d0dab87fee50e..77e791d5d162f5 100644 --- a/Mathlib/MeasureTheory/Measure/Trim.lean +++ b/Mathlib/MeasureTheory/Measure/Trim.lean @@ -10,7 +10,6 @@ public import Mathlib.MeasureTheory.Measure.Typeclasses.SFinite /-! # Restriction of a measure to a sub-σ-algebra - ## Main definitions * `MeasureTheory.Measure.trim`: restriction of a measure to a sub-sigma algebra. diff --git a/Mathlib/MeasureTheory/Measure/Typeclasses/ZeroOne.lean b/Mathlib/MeasureTheory/Measure/Typeclasses/ZeroOne.lean index fda22f4abeec32..c307bb6fe88982 100644 --- a/Mathlib/MeasureTheory/Measure/Typeclasses/ZeroOne.lean +++ b/Mathlib/MeasureTheory/Measure/Typeclasses/ZeroOne.lean @@ -10,7 +10,6 @@ public import Mathlib.MeasureTheory.Constructions.Polish.Basic public import Mathlib.MeasureTheory.Measure.Dirac /-! - We introduce the typeclass `IsZeroOneMeasure` for measures that only take the values `0` and `1`. ## Main definitions diff --git a/Mathlib/MeasureTheory/VectorMeasure/Basic.lean b/Mathlib/MeasureTheory/VectorMeasure/Basic.lean index 462e1339e6b361..55ca6903ec3ffd 100644 --- a/Mathlib/MeasureTheory/VectorMeasure/Basic.lean +++ b/Mathlib/MeasureTheory/VectorMeasure/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.MeasureTheory.Measure.Real public import Mathlib.Topology.Algebra.InfiniteSum.Module /-! - # Vector-valued measures This file defines vector-valued measures, which are σ-additive functions from a set to an diff --git a/Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean b/Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean index 1bbde10a8da904..65ded7f32a90be 100644 --- a/Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean +++ b/Mathlib/MeasureTheory/VectorMeasure/WithDensity.lean @@ -9,7 +9,6 @@ public import Mathlib.MeasureTheory.VectorMeasure.Basic public import Mathlib.MeasureTheory.Function.AEEqOfIntegral /-! - # Vector measure defined by an integral Given a measure `μ` and an integrable function `f : α → E`, we can define a vector measure `v` such diff --git a/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean b/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean index 7e2c2bc1ec3093..f7ab4c41798da9 100644 --- a/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean +++ b/Mathlib/ModelTheory/Algebra/Field/IsAlgClosed.lean @@ -12,7 +12,6 @@ public import Mathlib.ModelTheory.Algebra.Field.CharP public import Mathlib.ModelTheory.Satisfiability /-! - # The First-Order Theory of Algebraically Closed Fields This file defines the theory of algebraically closed fields of characteristic `p`, as well diff --git a/Mathlib/ModelTheory/Algebra/Ring/Definability.lean b/Mathlib/ModelTheory/Algebra/Ring/Definability.lean index bc64fe626ef1b1..578aa730037e75 100644 --- a/Mathlib/ModelTheory/Algebra/Ring/Definability.lean +++ b/Mathlib/ModelTheory/Algebra/Ring/Definability.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Nullstellensatz public import Mathlib.ModelTheory.Algebra.Ring.FreeCommRing /-! - # Definable Subsets in the language of rings This file proves that the set of zeros of a multivariable polynomial is a definable subset. diff --git a/Mathlib/ModelTheory/Complexity.lean b/Mathlib/ModelTheory/Complexity.lean index 88f2d680cc352f..086df43aabe461 100644 --- a/Mathlib/ModelTheory/Complexity.lean +++ b/Mathlib/ModelTheory/Complexity.lean @@ -22,7 +22,6 @@ This file defines quantifier complexity of first-order formulas, and constructs when it consists of a series of quantifiers applied to a quantifier-free formula. - `FirstOrder.Language.BoundedFormula.toPrenex` constructs a prenex normal form of a given formula. - ## Main Results - `FirstOrder.Language.BoundedFormula.realize_toPrenex` shows that the prenex normal form of a diff --git a/Mathlib/ModelTheory/FinitelyGenerated.lean b/Mathlib/ModelTheory/FinitelyGenerated.lean index d5341b1c86ea86..41aefbf94406f9 100644 --- a/Mathlib/ModelTheory/FinitelyGenerated.lean +++ b/Mathlib/ModelTheory/FinitelyGenerated.lean @@ -21,7 +21,6 @@ generated, similarly to other finitely-generated objects in the algebra library. - `FirstOrder.Language.Substructure.CG` indicates that a substructure is countably generated. - `FirstOrder.Language.Structure.CG` indicates that a structure is countably generated. - ## TODO Develop a more unified definition of finite generation using the theory of closure operators, or use diff --git a/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean b/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean index eaa68d6f06b3d3..f72c83fba0222e 100644 --- a/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean +++ b/Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.RootsOfUnity.Minpoly public import Mathlib.FieldTheory.KrullTopology /-! - # The cyclotomic character Let `L` be an integral domain and let `n : ℕ+` be a positive integer. If `μₙ` is the diff --git a/Mathlib/NumberTheory/Harmonic/Bounds.lean b/Mathlib/NumberTheory/Harmonic/Bounds.lean index 9bf769b83daeb5..07e8a81c59c7ab 100644 --- a/Mathlib/NumberTheory/Harmonic/Bounds.lean +++ b/Mathlib/NumberTheory/Harmonic/Bounds.lean @@ -10,7 +10,6 @@ public import Mathlib.Analysis.SumIntegralComparisons public import Mathlib.NumberTheory.Harmonic.Defs /-! - This file proves $\log(n + 1) \le H_n \le 1 + \log(n)$ for all natural numbers $n$. -/ diff --git a/Mathlib/NumberTheory/Harmonic/Defs.lean b/Mathlib/NumberTheory/Harmonic/Defs.lean index f35a54ee302524..7acb19642e2060 100644 --- a/Mathlib/NumberTheory/Harmonic/Defs.lean +++ b/Mathlib/NumberTheory/Harmonic/Defs.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Rat.Defs public import Mathlib.Algebra.BigOperators.Group.Finset.Basic /-! - This file defines the harmonic numbers. * `Mathlib/NumberTheory/Harmonic/Int.lean` proves that the `n`th harmonic number is not an integer. diff --git a/Mathlib/NumberTheory/Harmonic/Int.lean b/Mathlib/NumberTheory/Harmonic/Int.lean index 5c3847f0972813..38864f0fba8053 100644 --- a/Mathlib/NumberTheory/Harmonic/Int.lean +++ b/Mathlib/NumberTheory/Harmonic/Int.lean @@ -10,7 +10,6 @@ public import Mathlib.NumberTheory.Padics.PadicNumbers public import Mathlib.Tactic.Positivity /-! - The nth Harmonic number is not an integer. We formalize the proof using 2-adic valuations. This proof is due to Kürschák. diff --git a/Mathlib/NumberTheory/LocalField/Basic.lean b/Mathlib/NumberTheory/LocalField/Basic.lean index dd9694d4d63bb9..9949da1dc568e2 100644 --- a/Mathlib/NumberTheory/LocalField/Basic.lean +++ b/Mathlib/NumberTheory/LocalField/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Algebra.Valued.LocallyCompact public import Mathlib.Topology.Algebra.Valued.ValuativeRel /-! - # Definition of (Non-archimedean) local fields Given a topological field `K` equipped with an equivalence class of valuations (a `ValuativeRel`), diff --git a/Mathlib/NumberTheory/LucasLehmer.lean b/Mathlib/NumberTheory/LucasLehmer.lean index 137d17b82b52ba..ef3a21ac4e7f9e 100644 --- a/Mathlib/NumberTheory/LucasLehmer.lean +++ b/Mathlib/NumberTheory/LucasLehmer.lean @@ -17,7 +17,6 @@ prove `lucasLehmerResidue p = 0 ↔ Prime (mersenne p)`. We construct a `norm_num` extension to calculate this residue to certify primality of Mersenne primes using `lucas_lehmer_sufficiency`. - ## TODO - Speed up the calculations using `n ≡ (n % 2^p) + (n / 2^p) [MOD 2^p - 1]`. diff --git a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Transform.lean b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Transform.lean index f8f56c73da9b4a..7ee32d28162821 100644 --- a/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Transform.lean +++ b/Mathlib/NumberTheory/ModularForms/EisensteinSeries/E2/Transform.lean @@ -24,7 +24,6 @@ the action of `S = [[0, -1], [1, 0]]`. * **`G2_T_transform`**: `G2` is invariant under `T = [[1, 1], [0, 1]]`: `G₂|[2] T = G₂` - * **`G2_S_transform`**: The transformation under `S` involves a correction term: `G₂(z) = z⁻² G₂(-1/z) - 2πi/z` diff --git a/Mathlib/NumberTheory/NumberField/Discriminant/Different.lean b/Mathlib/NumberTheory/NumberField/Discriminant/Different.lean index a587564c84f2c8..35484414644253 100644 --- a/Mathlib/NumberTheory/NumberField/Discriminant/Different.lean +++ b/Mathlib/NumberTheory/NumberField/Discriminant/Different.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.DedekindDomain.LinearDisjoint public import Mathlib.RingTheory.Ideal.Norm.RelNorm /-! - # (Absolute) Discriminant and Different Ideal ## Main results diff --git a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean index 51748458774f32..cbdac1274a24fd 100644 --- a/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean +++ b/Mathlib/NumberTheory/NumberField/FractionalIdeal.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.FractionalIdeal.Norm public import Mathlib.RingTheory.FractionalIdeal.Operations /-! - # Fractional ideals of number fields Prove some results on the fractional ideals of number fields. diff --git a/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean b/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean index 5baa65ba634bb0..d0602af036ef3d 100644 --- a/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean +++ b/Mathlib/NumberTheory/NumberField/InfinitePlace/Basic.lean @@ -526,7 +526,6 @@ end IsPrimitiveRoot end NumberField.InfinitePlace /-! - ## The infinite place of the rationals. -/ diff --git a/Mathlib/NumberTheory/Padics/RingHoms.lean b/Mathlib/NumberTheory/Padics/RingHoms.lean index 7c697875793946..b45058d53c0abd 100644 --- a/Mathlib/NumberTheory/Padics/RingHoms.lean +++ b/Mathlib/NumberTheory/Padics/RingHoms.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.LocalRing.ResidueField.Defs public import Mathlib.RingTheory.ZMod /-! - # Relating `ℤ_[p]` to `ZMod (p ^ n)`, aka `ℤ/p^nℤ`. In this file we establish connections between the `p`-adic integers `ℤ_[p]` diff --git a/Mathlib/NumberTheory/RamificationInertia/HilbertTheory.lean b/Mathlib/NumberTheory/RamificationInertia/HilbertTheory.lean index b0714f550d861a..dea1f9d7231454 100644 --- a/Mathlib/NumberTheory/RamificationInertia/HilbertTheory.lean +++ b/Mathlib/NumberTheory/RamificationInertia/HilbertTheory.lean @@ -9,7 +9,6 @@ public import Mathlib.NumberTheory.RamificationInertia.Galois public import Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients /-! - # Decomposition and Inertia fields In this file, we develop Hilbert Theory on the splitting of prime ideals in a Galois extension. diff --git a/Mathlib/NumberTheory/RamificationInertia/Unramified.lean b/Mathlib/NumberTheory/RamificationInertia/Unramified.lean index c484d8c00d5481..a35b6d7d5e2210 100644 --- a/Mathlib/NumberTheory/RamificationInertia/Unramified.lean +++ b/Mathlib/NumberTheory/RamificationInertia/Unramified.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients public import Mathlib.RingTheory.RamificationInertia.Basic /-! - # Unramified and ramification index We connect `Ideal.ramificationIdx` to the commutative algebra notion predicate of `IsUnramifiedAt`. diff --git a/Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean b/Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean index 0cffb151d7bb9e..c9add76a8365ac 100644 --- a/Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean +++ b/Mathlib/NumberTheory/Transcendental/Liouville/Basic.lean @@ -13,7 +13,6 @@ public import Mathlib.Topology.Algebra.Polynomial import Mathlib.Algebra.Order.Interval.Set.Group /-! - # Liouville's theorem This file contains a proof of Liouville's theorem stating that all Liouville numbers are diff --git a/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleNumber.lean b/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleNumber.lean index 2242416cd8e260..c7d2f20949741d 100644 --- a/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleNumber.lean +++ b/Mathlib/NumberTheory/Transcendental/Liouville/LiouvilleNumber.lean @@ -8,7 +8,6 @@ module public import Mathlib.NumberTheory.Transcendental.Liouville.Basic /-! - # Liouville constants This file contains a construction of a family of Liouville numbers, indexed by a natural number $m$. diff --git a/Mathlib/Order/Bounds/Image.lean b/Mathlib/Order/Bounds/Image.lean index d019ab9910b1a5..0e40c70f29f845 100644 --- a/Mathlib/Order/Bounds/Image.lean +++ b/Mathlib/Order/Bounds/Image.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Set.NAry public import Mathlib.Order.Bounds.Basic /-! - # Images of upper/lower bounds under monotone functions In this file we prove various results about the behaviour of bounds under monotone/antitone maps. diff --git a/Mathlib/Order/Compare.lean b/Mathlib/Order/Compare.lean index 29da3037a99a1f..0beca71de3aecf 100644 --- a/Mathlib/Order/Compare.lean +++ b/Mathlib/Order/Compare.lean @@ -13,7 +13,6 @@ public import Mathlib.Order.OrderDual This file provides basic results about orderings and comparison in linear orders. - ## Definitions * `CmpLE`: An `Ordering` from `≤`. diff --git a/Mathlib/Order/Filter/Extr.lean b/Mathlib/Order/Filter/Extr.lean index f8337827d8f967..f0d43622eaea8c 100644 --- a/Mathlib/Order/Filter/Extr.lean +++ b/Mathlib/Order/Filter/Extr.lean @@ -58,7 +58,6 @@ Similar predicates with `on` suffix are particular cases for `l = 𝓟 s`. * `is**.max`, `is**.min`, `is**.sup`, `is**.inf` : similarly for `is**.add` for pointwise `max`, `min`, `sup`, `inf`, respectively. - ### Miscellaneous definitions * `is**_const` : any point is both a minimum and maximum for a constant function; diff --git a/Mathlib/Order/Height.lean b/Mathlib/Order/Height.lean index a3730fbc6acb25..ebfe8a9ac7bd89 100644 --- a/Mathlib/Order/Height.lean +++ b/Mathlib/Order/Height.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.ENat.Lattice public import Mathlib.Data.Set.Card /-! - # Maximal length of chains This file contains lemmas to work with the maximal lengths of chains of arbitrary relations. See diff --git a/Mathlib/Order/PropInstances.lean b/Mathlib/Order/PropInstances.lean index 9437e3a7f57aae..262ac5dea4da80 100644 --- a/Mathlib/Order/PropInstances.lean +++ b/Mathlib/Order/PropInstances.lean @@ -8,7 +8,6 @@ module public import Mathlib.Order.Disjoint /-! - # The order on `Prop` Instances on `Prop` such as `DistribLattice`, `BoundedOrder`, `LinearOrder`. diff --git a/Mathlib/Order/ScottContinuity/Complete.lean b/Mathlib/Order/ScottContinuity/Complete.lean index fbb3fa352c3e69..05a7eae4da1371 100644 --- a/Mathlib/Order/ScottContinuity/Complete.lean +++ b/Mathlib/Order/ScottContinuity/Complete.lean @@ -8,7 +8,6 @@ module public import Mathlib.Order.ScottContinuity.Prod /-! - # Scott continuity on complete lattices ## Main results diff --git a/Mathlib/Order/SuccPred/CompleteLinearOrder.lean b/Mathlib/Order/SuccPred/CompleteLinearOrder.lean index 10c2c2e1f76a4b..b55e631c838564 100644 --- a/Mathlib/Order/SuccPred/CompleteLinearOrder.lean +++ b/Mathlib/Order/SuccPred/CompleteLinearOrder.lean @@ -9,7 +9,6 @@ public import Mathlib.Order.ConditionallyCompleteLattice.Indexed public import Mathlib.Order.SuccPred.Limit /-! - # Relation between `IsSuccPrelimit` and `iSup` in (conditionally) complete linear orders. -/ diff --git a/Mathlib/Order/Types/Arithmetic.lean b/Mathlib/Order/Types/Arithmetic.lean index fd560d56d1bd7e..0b140439a614ca 100644 --- a/Mathlib/Order/Types/Arithmetic.lean +++ b/Mathlib/Order/Types/Arithmetic.lean @@ -10,7 +10,6 @@ public import Mathlib.Order.Types.Defs public import Mathlib.SetTheory.Cardinal.Order /-! - ## Main definitions * `OrderType.card o`: the cardinality of an OrderType `o`. diff --git a/Mathlib/Probability/BorelCantelli.lean b/Mathlib/Probability/BorelCantelli.lean index 8fd7476d2ae477..7a434de6d4a8a1 100644 --- a/Mathlib/Probability/BorelCantelli.lean +++ b/Mathlib/Probability/BorelCantelli.lean @@ -10,7 +10,6 @@ public import Mathlib.Probability.ConditionalExpectation public import Mathlib.Probability.Independence.Basic /-! - # The second Borel-Cantelli lemma This file contains the *second Borel-Cantelli lemma* which states that, given a sequence of diff --git a/Mathlib/Probability/ConditionalExpectation.lean b/Mathlib/Probability/ConditionalExpectation.lean index 1e03e0357cec29..9b2630eb23df37 100644 --- a/Mathlib/Probability/ConditionalExpectation.lean +++ b/Mathlib/Probability/ConditionalExpectation.lean @@ -10,7 +10,6 @@ public import Mathlib.Probability.Independence.Basic public import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic /-! - # Probabilistic properties of the conditional expectation This file contains some properties about the conditional expectation which does not belong in diff --git a/Mathlib/Probability/Independence/Process/HasIndepIncrements/IsGaussianProcess.lean b/Mathlib/Probability/Independence/Process/HasIndepIncrements/IsGaussianProcess.lean index 83a680adb52220..1d83e1cb7a7788 100644 --- a/Mathlib/Probability/Independence/Process/HasIndepIncrements/IsGaussianProcess.lean +++ b/Mathlib/Probability/Independence/Process/HasIndepIncrements/IsGaussianProcess.lean @@ -12,7 +12,6 @@ import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Basic import Mathlib.Probability.Distributions.Gaussian.HasGaussianLaw.Independence /-! - # A stochastic process with independent increments and Gaussian marginals is Gaussian We prove that a stochastic process with independent increments and Gaussian marginals is Gaussian. diff --git a/Mathlib/Probability/Kernel/Composition/ParallelComp.lean b/Mathlib/Probability/Kernel/Composition/ParallelComp.lean index 37945ad527781c..aceb966a35974a 100644 --- a/Mathlib/Probability/Kernel/Composition/ParallelComp.lean +++ b/Mathlib/Probability/Kernel/Composition/ParallelComp.lean @@ -10,7 +10,6 @@ public import Mathlib.Probability.Kernel.Composition.MapComap public import Mathlib.Probability.Kernel.MeasurableLIntegral /-! - # Parallel composition of kernels Two kernels `κ : Kernel α β` and `η : Kernel γ δ` can be applied in parallel to give a kernel diff --git a/Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean b/Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean index acb68911b7da96..ad0bf46b443d44 100644 --- a/Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean +++ b/Mathlib/Probability/Kernel/IonescuTulcea/Traj.lean @@ -60,7 +60,6 @@ expectation. * `condDistrib_trajMeasure`: a regular conditional probability distribution of the point at time `a + 1` given the trajectory up to time `a` corresponds to the kernel `κ a`. - ## Implementation notes The kernel `traj κ a` is built using the Carathéodory extension theorem. First we build a projective diff --git a/Mathlib/Probability/Kernel/Posterior.lean b/Mathlib/Probability/Kernel/Posterior.lean index 78f2c8c4592cbd..16b6b9c51abbab 100644 --- a/Mathlib/Probability/Kernel/Posterior.lean +++ b/Mathlib/Probability/Kernel/Posterior.lean @@ -11,7 +11,6 @@ public import Mathlib.Probability.Kernel.Disintegration.StandardBorel public import Mathlib.Probability.Kernel.Deterministic /-! - # Posterior kernel For `μ : Measure Ω` (called prior measure), seen as a measure on a parameter, and a kernel diff --git a/Mathlib/Probability/Martingale/BorelCantelli.lean b/Mathlib/Probability/Martingale/BorelCantelli.lean index ae70ff31aa470b..a8b3a3f717dc19 100644 --- a/Mathlib/Probability/Martingale/BorelCantelli.lean +++ b/Mathlib/Probability/Martingale/BorelCantelli.lean @@ -11,7 +11,6 @@ public import Mathlib.Probability.Martingale.Convergence public import Mathlib.Probability.Martingale.OptionalStopping /-! - # Generalized Borel-Cantelli lemma This file proves Lévy's generalized Borel-Cantelli lemma which is a generalization of the diff --git a/Mathlib/Probability/Martingale/Convergence.lean b/Mathlib/Probability/Martingale/Convergence.lean index f304cfd1f359f9..970c888240791f 100644 --- a/Mathlib/Probability/Martingale/Convergence.lean +++ b/Mathlib/Probability/Martingale/Convergence.lean @@ -10,7 +10,6 @@ public import Mathlib.MeasureTheory.Function.UniformIntegrable public import Mathlib.Probability.Martingale.Upcrossing /-! - # Martingale convergence theorems The martingale convergence theorems are a collection of theorems characterizing the convergence @@ -58,7 +57,6 @@ variable {a b : ℝ} {f : ℕ → Ω → ℝ} {ω : Ω} {R : ℝ≥0} section AeConvergence /-! - ### Almost everywhere martingale convergence theorem We will now prove the almost everywhere martingale convergence theorem. @@ -243,7 +241,6 @@ section L1Convergence variable [IsFiniteMeasure μ] {g : Ω → ℝ} /-! - ### L¹ martingale convergence theorem We will now prove the L¹ martingale convergence theorems. diff --git a/Mathlib/Probability/Martingale/Upcrossing.lean b/Mathlib/Probability/Martingale/Upcrossing.lean index d266f8502d70b2..14b70e90ba6705 100644 --- a/Mathlib/Probability/Martingale/Upcrossing.lean +++ b/Mathlib/Probability/Martingale/Upcrossing.lean @@ -12,7 +12,6 @@ public import Mathlib.Probability.Martingale.Basic public import Mathlib.Tactic.AdaptationNote /-! - # Doob's upcrossing estimate Given a discrete real-valued submartingale $(f_n)_{n \in \mathbb{N}}$, denoting by $U_N(a, b)$ the @@ -69,7 +68,6 @@ namespace MeasureTheory variable {Ω ι : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω} /-! - ## Proof outline In this section, we will denote by $U_N(a, b)$ the number of upcrossings of $(f_n)$ from below $a$ @@ -697,7 +695,6 @@ theorem Submartingale.mul_integral_upcrossingsBefore_le_integral_pos_part [IsFin (integral_nonneg fun ω => posPart_nonneg _) /-! - ### Variant of the upcrossing estimate Now, we would like to prove a variant of the upcrossing estimate obtained by taking the supremum diff --git a/Mathlib/RepresentationTheory/Action.lean b/Mathlib/RepresentationTheory/Action.lean index e756bfb73b9521..06e5e247164d91 100644 --- a/Mathlib/RepresentationTheory/Action.lean +++ b/Mathlib/RepresentationTheory/Action.lean @@ -10,7 +10,6 @@ public import Mathlib.RepresentationTheory.Intertwining public import Mathlib.RingTheory.TensorProduct.MonoidAlgebra /-! - ## Main Purpose This file is the preliminary for the `linearize` functor from `Action (Type w) G` to `Rep k G`, constructing the functor from the `Representation` would reduce the amount of DefEq abuses that we diff --git a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean index ff051e312747b9..d3043ff95867a3 100644 --- a/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean +++ b/Mathlib/RingTheory/AdicCompletion/AsTensorProduct.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.AdicCompletion.Exactness public import Mathlib.RingTheory.Flat.Tensor /-! - # Adic completion as tensor product In this file we examine properties of the natural map @@ -219,7 +218,6 @@ variable {R : Type u} [CommRing R] (I : Ideal R) variable (M : Type u) [AddCommGroup M] [Module R M] /-! - ### Noetherian case Suppose `R` is Noetherian. Then we show that the canonical map diff --git a/Mathlib/RingTheory/AdicCompletion/Topology.lean b/Mathlib/RingTheory/AdicCompletion/Topology.lean index e0c6c0d33a17c6..780296f8ecb131 100644 --- a/Mathlib/RingTheory/AdicCompletion/Topology.lean +++ b/Mathlib/RingTheory/AdicCompletion/Topology.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.AdicCompletion.Basic public import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology /-! - # Connection between adic properties and topological properties ## Main results diff --git a/Mathlib/RingTheory/Bezout.lean b/Mathlib/RingTheory/Bezout.lean index f3674690560a5f..a47b517655fee9 100644 --- a/Mathlib/RingTheory/Bezout.lean +++ b/Mathlib/RingTheory/Bezout.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.PrincipalIdealDomain /-! - # Bézout rings A Bézout ring (Bezout ring) is a ring whose finitely generated ideals are principal. diff --git a/Mathlib/RingTheory/ChainOfDivisors.lean b/Mathlib/RingTheory/ChainOfDivisors.lean index e9e83174dbeb93..29b360dcb395e6 100644 --- a/Mathlib/RingTheory/ChainOfDivisors.lean +++ b/Mathlib/RingTheory/ChainOfDivisors.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity public import Mathlib.Order.Atoms public import Mathlib.Order.Hom.Bounded /-! - # Chains of divisors The results in this file show that in the monoid `Associates M` of a `UniqueFactorizationMonoid` @@ -29,7 +28,6 @@ and the set of factors of `a`. between the set of factors of `a : M` and `b : N` then for any prime `p ∣ a`, `multiplicity p a = multiplicity (d p) b` - ## TODO - Create a structure for chains of divisors. - Simplify proof of `mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors` using diff --git a/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean b/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean index f1a75c850f1310..bfa8746276817c 100644 --- a/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean +++ b/Mathlib/RingTheory/DiscreteValuationRing/TFAE.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Ideal.Cotangent public import Mathlib.RingTheory.KrullDimension.Zero /-! - # Equivalent conditions for DVR In `IsDiscreteValuationRing.TFAE`, we show that the following are equivalent for a diff --git a/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean b/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean index c36b5a97e98dfb..5acd2ca6007f05 100644 --- a/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean +++ b/Mathlib/RingTheory/DividedPowerAlgebra/Init.lean @@ -62,7 +62,6 @@ divided powers (`DividedPowerAlgebra.lift`). * Show in upcoming files that `DividedPowerAlgebra R M` has divided powers. - -/ @[expose] public section diff --git a/Mathlib/RingTheory/Etale/Basic.lean b/Mathlib/RingTheory/Etale/Basic.lean index c941952589073a..d8aab8f19cfd07 100644 --- a/Mathlib/RingTheory/Etale/Basic.lean +++ b/Mathlib/RingTheory/Etale/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Smooth.Basic public import Mathlib.RingTheory.Unramified.Basic /-! - # Étale morphisms An `R`-algebra `A` is formally etale if `Ω[A⁄R]` and `H¹(L_{A/R})` both vanish. @@ -159,7 +158,6 @@ end BaseChange section Localization /-! - We now consider a commutative square of commutative rings ``` diff --git a/Mathlib/RingTheory/Etale/Pi.lean b/Mathlib/RingTheory/Etale/Pi.lean index aaf20e477d5d52..f03942b63d0b65 100644 --- a/Mathlib/RingTheory/Etale/Pi.lean +++ b/Mathlib/RingTheory/Etale/Pi.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Etale.Basic public import Mathlib.RingTheory.Finiteness.FinitePresentationLocal /-! - # Formal-étaleness of finite products of rings ## Main result diff --git a/Mathlib/RingTheory/Etale/StandardEtale.lean b/Mathlib/RingTheory/Etale/StandardEtale.lean index 64f44f09d4c17b..e103cd59120cb9 100644 --- a/Mathlib/RingTheory/Etale/StandardEtale.lean +++ b/Mathlib/RingTheory/Etale/StandardEtale.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Extension.Presentation.Submersive public import Mathlib.RingTheory.Ideal.IdempotentFG /-! - # Standard etale maps ## Main definitions diff --git a/Mathlib/RingTheory/Extension/Basic.lean b/Mathlib/RingTheory/Extension/Basic.lean index 949acefa98bf7c..14431755100d29 100644 --- a/Mathlib/RingTheory/Extension/Basic.lean +++ b/Mathlib/RingTheory/Extension/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.Cotangent public import Mathlib.RingTheory.Localization.Defs /-! - # Extension of algebras ## Main definitions diff --git a/Mathlib/RingTheory/Extension/Cotangent/Basic.lean b/Mathlib/RingTheory/Extension/Cotangent/Basic.lean index 0f6e4172bbc0d5..dcad16cf2ec8ee 100644 --- a/Mathlib/RingTheory/Extension/Cotangent/Basic.lean +++ b/Mathlib/RingTheory/Extension/Cotangent/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Module.FinitePresentation public import Mathlib.RingTheory.Extension.Presentation.Basic /-! - # Naive cotangent complex associated to a presentation. Given a presentation `0 → I → R[x₁,...,xₙ] → S → 0` (or equivalently a closed embedding `S ↪ Aⁿ` diff --git a/Mathlib/RingTheory/Extension/Generators.lean b/Mathlib/RingTheory/Extension/Generators.lean index 5c03910ea8864e..d9495030c09386 100644 --- a/Mathlib/RingTheory/Extension/Generators.lean +++ b/Mathlib/RingTheory/Extension/Generators.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.TensorProduct.MvPolynomial public import Mathlib.RingTheory.Extension.Basic /-! - # Generators of algebras ## Main definition diff --git a/Mathlib/RingTheory/Extension/Presentation/Basic.lean b/Mathlib/RingTheory/Extension/Presentation/Basic.lean index 5dc88a6f3a2fbf..2dfb9a16537e17 100644 --- a/Mathlib/RingTheory/Extension/Presentation/Basic.lean +++ b/Mathlib/RingTheory/Extension/Presentation/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.MvPolynomial.Localization public import Mathlib.RingTheory.TensorProduct.MvPolynomial /-! - # Presentations of algebras A presentation of an `R`-algebra `S` is a distinguished family of generators and relations. diff --git a/Mathlib/RingTheory/Filtration.lean b/Mathlib/RingTheory/Filtration.lean index f909cc28ab729a..29f26342fcb4e0 100644 --- a/Mathlib/RingTheory/Filtration.lean +++ b/Mathlib/RingTheory/Filtration.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic public import Mathlib.RingTheory.ReesAlgebra /-! - # `I`-filtrations of modules This file contains the definitions and basic results around (stable) `I`-filtrations of modules. diff --git a/Mathlib/RingTheory/FiniteStability.lean b/Mathlib/RingTheory/FiniteStability.lean index 50db1a6af15029..05801544ffdd0e 100644 --- a/Mathlib/RingTheory/FiniteStability.lean +++ b/Mathlib/RingTheory/FiniteStability.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.FinitePresentation public import Mathlib.RingTheory.TensorProduct.MvPolynomial /-! - # Stability of finiteness conditions in commutative algebra In this file we show that `Algebra.FiniteType` and `Algebra.FinitePresentation` are diff --git a/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean b/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean index 0b759d65bcc3dc..236d4f878f8159 100644 --- a/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean +++ b/Mathlib/RingTheory/Finiteness/FinitePresentationLocal.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Finiteness.FiniteTypeLocal public import Mathlib.RingTheory.Localization.Away.AdjoinRoot /-! - # `Algebra.FinitePresentation` is local In this file we show that being a finitely presented algebra is local. diff --git a/Mathlib/RingTheory/Finiteness/FiniteTypeLocal.lean b/Mathlib/RingTheory/Finiteness/FiniteTypeLocal.lean index f41af9b621f778..55b83571c93b3f 100644 --- a/Mathlib/RingTheory/Finiteness/FiniteTypeLocal.lean +++ b/Mathlib/RingTheory/Finiteness/FiniteTypeLocal.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Localization.Finiteness public import Mathlib.RingTheory.Localization.BaseChange /-! - # Locality of `Algebra.FiniteType` In this file we show that finite-type is local on the source and the target. diff --git a/Mathlib/RingTheory/Flat/IsBaseChange.lean b/Mathlib/RingTheory/Flat/IsBaseChange.lean index 0697065de7dc26..457ea663a9533b 100644 --- a/Mathlib/RingTheory/Flat/IsBaseChange.lean +++ b/Mathlib/RingTheory/Flat/IsBaseChange.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.FiveLemma public import Mathlib.RingTheory.Flat.Basic /-! - # Lemmas about `IsBaseChange` under exact sequences In this file, we show that if `S` is a flat `R`-algebra, taking kernels commutes with base change diff --git a/Mathlib/RingTheory/Flat/Rank.lean b/Mathlib/RingTheory/Flat/Rank.lean index 7d35764a596f6b..4a4779ca7ac8ee 100644 --- a/Mathlib/RingTheory/Flat/Rank.lean +++ b/Mathlib/RingTheory/Flat/Rank.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.FreeLocus public import Mathlib.RingTheory.RingHom.Flat /-! - # Results for the rank of a finite flat algebra In this file we study a finite, flat `R`-algebra `S` and relate injectivity and diff --git a/Mathlib/RingTheory/FractionalIdeal/Norm.lean b/Mathlib/RingTheory/FractionalIdeal/Norm.lean index 69a97d0de75ad2..7aba241f09cb7c 100644 --- a/Mathlib/RingTheory/FractionalIdeal/Norm.lean +++ b/Mathlib/RingTheory/FractionalIdeal/Norm.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.Norm.AbsNorm public import Mathlib.RingTheory.Localization.NormTrace /-! - # Fractional ideal norms This file defines the absolute ideal norm of a fractional ideal `I : FractionalIdeal R⁰ K` where diff --git a/Mathlib/RingTheory/GradedAlgebra/FiniteType.lean b/Mathlib/RingTheory/GradedAlgebra/FiniteType.lean index 8a9a01a52288dc..16ad798fd48f6e 100644 --- a/Mathlib/RingTheory/GradedAlgebra/FiniteType.lean +++ b/Mathlib/RingTheory/GradedAlgebra/FiniteType.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.FiniteType public import Mathlib.RingTheory.GradedAlgebra.Basic /-! - # Graded rings of finite type We show that graded rings of finite type (over the 0-th component) are diff --git a/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean b/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean index 4658ca2c7240e3..3a8f4ea615d5e5 100644 --- a/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean +++ b/Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean @@ -69,7 +69,6 @@ circumvent this, we quotient `NumDenSameDeg 𝒜 x` by the kernel of `c ↦ c.nu * [Robin Hartshorne, *Algebraic Geometry*][Har77] - -/ @[expose] public section diff --git a/Mathlib/RingTheory/GradedAlgebra/Radical.lean b/Mathlib/RingTheory/GradedAlgebra/Radical.lean index e83178cf7ff08f..26be026e890c51 100644 --- a/Mathlib/RingTheory/GradedAlgebra/Radical.lean +++ b/Mathlib/RingTheory/GradedAlgebra/Radical.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal /-! - This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal ## Main statements diff --git a/Mathlib/RingTheory/HahnSeries/HahnEmbedding.lean b/Mathlib/RingTheory/HahnSeries/HahnEmbedding.lean index 142a3a1712b3b4..b996e2cbd10ca8 100644 --- a/Mathlib/RingTheory/HahnSeries/HahnEmbedding.lean +++ b/Mathlib/RingTheory/HahnSeries/HahnEmbedding.lean @@ -13,7 +13,6 @@ public import Mathlib.Data.Real.Embedding public import Mathlib.GroupTheory.DivisibleHull /-! - # Hahn embedding theorem In this file, we prove the Hahn embedding theorem: every linearly ordered abelian group diff --git a/Mathlib/RingTheory/HahnSeries/Lex.lean b/Mathlib/RingTheory/HahnSeries/Lex.lean index bd86c3046f4539..e4a75b6ae37e09 100644 --- a/Mathlib/RingTheory/HahnSeries/Lex.lean +++ b/Mathlib/RingTheory/HahnSeries/Lex.lean @@ -12,7 +12,6 @@ public import Mathlib.Order.PiLex public import Mathlib.RingTheory.HahnSeries.Multiplication /-! - # Lexicographical order on Hahn series In this file, we define lexicographical ordered `Lex R⟦Γ⟧`, and show this is a `LinearOrder` when diff --git a/Mathlib/RingTheory/HopfAlgebra/Basic.lean b/Mathlib/RingTheory/HopfAlgebra/Basic.lean index 7d10e13158862c..64288a659abb97 100644 --- a/Mathlib/RingTheory/HopfAlgebra/Basic.lean +++ b/Mathlib/RingTheory/HopfAlgebra/Basic.lean @@ -47,7 +47,6 @@ so we could deduce the facts here from an equivalence `HopfAlgCat R ≌ Hopf (Mo * [C. Kassel, *Quantum Groups* (§III.3)][Kassel1995] - -/ public section diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Basic.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Basic.lean index 4643bb3748d6f5..f9a4e9c76588d1 100644 --- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Basic.lean +++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian public import Mathlib.RingTheory.Noetherian.Basic /-! - # Associated primes of a module We provide the definition and related lemmas about associated primes of modules. diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean index 8981a4233c3901..8403e74641edf1 100644 --- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean +++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Finiteness.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.Defs public import Mathlib.RingTheory.Spectrum.Maximal.Basic /-! - # Finitely generated module over Noetherian ring have finitely many associated primes. In this file we proved that any finitely generated module over a Noetherian ring have finitely many diff --git a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean index 91c6a9966c74c6..aad5b818596bee 100644 --- a/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean +++ b/Mathlib/RingTheory/Ideal/AssociatedPrime/Localization.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic public import Mathlib.RingTheory.Support /-! - # Associated primes of localized module This file mainly proves the relation between `Ass(S⁻¹M)` and `Ass(M)` diff --git a/Mathlib/RingTheory/Ideal/Basic.lean b/Mathlib/RingTheory/Ideal/Basic.lean index 4ddfab679349ff..145f6570c77cb6 100644 --- a/Mathlib/RingTheory/Ideal/Basic.lean +++ b/Mathlib/RingTheory/Ideal/Basic.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Ideal.Maximal public import Mathlib.Tactic.FinCases /-! - # Ideals over a ring This file contains an assortment of definitions and results for `Ideal R`, diff --git a/Mathlib/RingTheory/Ideal/BigOperators.lean b/Mathlib/RingTheory/Ideal/BigOperators.lean index b057249c445035..54874dbd654a19 100644 --- a/Mathlib/RingTheory/Ideal/BigOperators.lean +++ b/Mathlib/RingTheory/Ideal/BigOperators.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Module.Submodule.Basic public import Mathlib.RingTheory.Ideal.Defs /-! - # Big operators for ideals This contains some results on the big operators `∑` and `∏` interacting with the `Ideal` type. diff --git a/Mathlib/RingTheory/Ideal/Defs.lean b/Mathlib/RingTheory/Ideal/Defs.lean index aee2dc55cc58e9..167e83f429081b 100644 --- a/Mathlib/RingTheory/Ideal/Defs.lean +++ b/Mathlib/RingTheory/Ideal/Defs.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Module.Submodule.Defs public import Mathlib.Tactic.Abel /-! - # Ideals over a ring This file defines `Ideal R`, the type of (left) ideals over a ring `R`. diff --git a/Mathlib/RingTheory/Ideal/Maximal.lean b/Mathlib/RingTheory/Ideal/Maximal.lean index f4cfefe052d189..21d79b70210cc6 100644 --- a/Mathlib/RingTheory/Ideal/Maximal.lean +++ b/Mathlib/RingTheory/Ideal/Maximal.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Ideal.Prime public import Mathlib.RingTheory.Ideal.Span /-! - # Ideals over a ring This file contains an assortment of definitions and results for `Ideal R`, diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean index 23e037a198388a..0b42ab922cb902 100644 --- a/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean +++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.Over public import Mathlib.Order.Minimal /-! - # Minimal primes We provide various results concerning the minimal primes above an ideal. diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean index d7db15b1f87253..295b533c46bccd 100644 --- a/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean +++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Colon.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian public import Mathlib.RingTheory.Noetherian.Basic /-! - # Minimal primes over a colon ideal We prove that a minimal prime over an ideal of the form `N.colon {x}` in a Noetherian ring is diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean index 0815cca79f45dc..87ad2e30a84c67 100644 --- a/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean +++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Localization.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Ideal.MinimalPrime.Basic public import Mathlib.RingTheory.Localization.AtPrime.Basic /-! - # Minimal primes and localization We provide various results concerning the minimal primes above an ideal that require the theory diff --git a/Mathlib/RingTheory/Ideal/MinimalPrime/Noetherian.lean b/Mathlib/RingTheory/Ideal/MinimalPrime/Noetherian.lean index 8863003728235c..71ee427a7d0b3f 100644 --- a/Mathlib/RingTheory/Ideal/MinimalPrime/Noetherian.lean +++ b/Mathlib/RingTheory/Ideal/MinimalPrime/Noetherian.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.MinimalPrime.Basic public import Mathlib.RingTheory.Noetherian.Defs /-! - # Finiteness of minimal primes We prove finiteness of minimal primes above an ideal. diff --git a/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean b/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean index a8f308649071e7..c14b96462ca721 100644 --- a/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean +++ b/Mathlib/RingTheory/Ideal/Norm/AbsNorm.lean @@ -16,7 +16,6 @@ public import Mathlib.RingTheory.Norm.Basic public import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicative /-! - # Ideal norms This file defines the absolute ideal norm `Ideal.absNorm (I : Ideal R) : ℕ` as the cardinality of diff --git a/Mathlib/RingTheory/Ideal/Norm/RelNorm.lean b/Mathlib/RingTheory/Ideal/Norm/RelNorm.lean index 829c6ed62493a2..8ac929f4c52627 100644 --- a/Mathlib/RingTheory/Ideal/Norm/RelNorm.lean +++ b/Mathlib/RingTheory/Ideal/Norm/RelNorm.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Ideal.Int public import Mathlib.RingTheory.NormalClosure /-! - # Ideal norms This file defines the relative ideal norm `Ideal.spanNorm R (I : Ideal S) : Ideal S` as the ideal diff --git a/Mathlib/RingTheory/Ideal/Prime.lean b/Mathlib/RingTheory/Ideal/Prime.lean index c19bef639068a2..8c6a5d2224fb25 100644 --- a/Mathlib/RingTheory/Ideal/Prime.lean +++ b/Mathlib/RingTheory/Ideal/Prime.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.GroupWithZero.NonZeroDivisors public import Mathlib.RingTheory.Ideal.Lattice /-! - # Prime ideals This file contains the definition of `Ideal.IsPrime` for prime ideals. diff --git a/Mathlib/RingTheory/Idempotents.lean b/Mathlib/RingTheory/Idempotents.lean index 815c057d4267b7..dd328e1ce7a939 100644 --- a/Mathlib/RingTheory/Idempotents.lean +++ b/Mathlib/RingTheory/Idempotents.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Ideal.Quotient.Operations public import Mathlib.Tactic.LinearCombination /-! - # Idempotents in rings The predicate `IsIdempotentElem` is defined for general monoids in diff --git a/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean b/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean index 554bb37bc74460..82d90d3b57959f 100644 --- a/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean +++ b/Mathlib/RingTheory/IntegralClosure/Algebra/Ideal.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.IntegralClosure.Algebra.Basic /-! - # Integrality over ideals ## Main results diff --git a/Mathlib/RingTheory/IntegralClosure/GoingDown.lean b/Mathlib/RingTheory/IntegralClosure/GoingDown.lean index a42ee0f39c77a2..2be42cf0a08cee 100644 --- a/Mathlib/RingTheory/IntegralClosure/GoingDown.lean +++ b/Mathlib/RingTheory/IntegralClosure/GoingDown.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.GoingDown public import Mathlib.RingTheory.IntegralClosure.Algebra.Ideal /-! - # Going down for integrally closed domains In this file, we provide the instance that any integral extension of `R ⊆ S` satisfies going down diff --git a/Mathlib/RingTheory/Kaehler/JacobiZariski.lean b/Mathlib/RingTheory/Kaehler/JacobiZariski.lean index b0793212737291..11e61b40d5aa18 100644 --- a/Mathlib/RingTheory/Kaehler/JacobiZariski.lean +++ b/Mathlib/RingTheory/Kaehler/JacobiZariski.lean @@ -11,7 +11,6 @@ public import Mathlib.Algebra.Module.SnakeLemma public import Mathlib.RingTheory.Flat.Basic /-! - # The Jacobi-Zariski exact sequence Given algebras $R \to S \to T$, the Jacobi-Zariski exact sequence is a long exact sequence diff --git a/Mathlib/RingTheory/KrullDimension/Module.lean b/Mathlib/RingTheory/KrullDimension/Module.lean index 677d938ca2ef8b..affb28d04938c8 100644 --- a/Mathlib/RingTheory/KrullDimension/Module.lean +++ b/Mathlib/RingTheory/KrullDimension/Module.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.KrullDimension.NonZeroDivisors public import Mathlib.RingTheory.Spectrum.Prime.Module /-! - # Krull Dimension of Module In this file we define `Module.supportDim R M` for an `R`-module `M` as diff --git a/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean b/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean index 02ba2e127fcead..10c73374bf84fc 100644 --- a/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean +++ b/Mathlib/RingTheory/KrullDimension/NonZeroDivisors.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.RingHom public import Mathlib.Algebra.MvPolynomial.CommRing /-! - # Krull dimension and non-zero-divisors ## Main results diff --git a/Mathlib/RingTheory/KrullDimension/Regular.lean b/Mathlib/RingTheory/KrullDimension/Regular.lean index 28d87d1a5cc7ca..1c5ea3fbf1dd5f 100644 --- a/Mathlib/RingTheory/KrullDimension/Regular.lean +++ b/Mathlib/RingTheory/KrullDimension/Regular.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Regular.RegularSequence public import Mathlib.RingTheory.Spectrum.Prime.LTSeries /-! - # Krull Dimension of quotient regular sequence ## Main results diff --git a/Mathlib/RingTheory/KrullDimension/Zero.lean b/Mathlib/RingTheory/KrullDimension/Zero.lean index 1971d6da9a135e..cd8bab99ea8b1e 100644 --- a/Mathlib/RingTheory/KrullDimension/Zero.lean +++ b/Mathlib/RingTheory/KrullDimension/Zero.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Jacobson.Ring public import Mathlib.RingTheory.Spectrum.Prime.Topology /-! - # Zero-dimensional rings We provide further API for zero-dimensional rings. diff --git a/Mathlib/RingTheory/Length.lean b/Mathlib/RingTheory/Length.lean index 002d65e28c0f45..e7a83eeab3a88d 100644 --- a/Mathlib/RingTheory/Length.lean +++ b/Mathlib/RingTheory/Length.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.FiniteLength public import Mathlib.LinearAlgebra.Dimension.Free /-! - # Length of modules ## Main results diff --git a/Mathlib/RingTheory/LinearDisjoint.lean b/Mathlib/RingTheory/LinearDisjoint.lean index fe0d53ff6a84e6..89da9aba046930 100644 --- a/Mathlib/RingTheory/LinearDisjoint.lean +++ b/Mathlib/RingTheory/LinearDisjoint.lean @@ -19,7 +19,6 @@ public import Mathlib.RingTheory.TensorProduct.Nontrivial public import Mathlib.RingTheory.Trace.Defs /-! - # Linearly disjoint subalgebras This file contains basics about linearly disjoint subalgebras. diff --git a/Mathlib/RingTheory/LocalProperties/Injective.lean b/Mathlib/RingTheory/LocalProperties/Injective.lean index 209bc9413acae2..e48b5b04498caf 100644 --- a/Mathlib/RingTheory/LocalProperties/Injective.lean +++ b/Mathlib/RingTheory/LocalProperties/Injective.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.LocalProperties.Basic public import Mathlib.RingTheory.LocalProperties.Exactness /-! - # Being injective is a local property ## Main Results diff --git a/Mathlib/RingTheory/LocalProperties/InjectiveDimension.lean b/Mathlib/RingTheory/LocalProperties/InjectiveDimension.lean index 3165b96a864fba..36753d1bb17b42 100644 --- a/Mathlib/RingTheory/LocalProperties/InjectiveDimension.lean +++ b/Mathlib/RingTheory/LocalProperties/InjectiveDimension.lean @@ -18,7 +18,6 @@ public import Mathlib.LinearAlgebra.Dimension.Finite public import Mathlib.RingTheory.LocalProperties.Injective /-! - # Relation of Injective Dimension with Localizations -/ diff --git a/Mathlib/RingTheory/LocalProperties/Projective.lean b/Mathlib/RingTheory/LocalProperties/Projective.lean index 668948e2ba7be2..42dffc3bdbf6b3 100644 --- a/Mathlib/RingTheory/LocalProperties/Projective.lean +++ b/Mathlib/RingTheory/LocalProperties/Projective.lean @@ -12,7 +12,6 @@ public import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition public import Mathlib.RingTheory.LocalProperties.Submodule /-! - # Being projective is a local property ## Main results diff --git a/Mathlib/RingTheory/LocalRing/Basic.lean b/Mathlib/RingTheory/LocalRing/Basic.lean index 63430eac95cd99..2a5b714b8caefc 100644 --- a/Mathlib/RingTheory/LocalRing/Basic.lean +++ b/Mathlib/RingTheory/LocalRing/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.LocalRing.Defs public import Mathlib.RingTheory.Ideal.Nonunits /-! - # Local rings We prove basic properties of local rings. diff --git a/Mathlib/RingTheory/LocalRing/Defs.lean b/Mathlib/RingTheory/LocalRing/Defs.lean index 9538400b1f4589..591f6dcc6fa945 100644 --- a/Mathlib/RingTheory/LocalRing/Defs.lean +++ b/Mathlib/RingTheory/LocalRing/Defs.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Group.Units.Defs public import Mathlib.Algebra.Ring.Defs /-! - # Local rings Define the notion of a local ring for non-commutative semirings. In the commutative case, diff --git a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Basic.lean b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Basic.lean index b1e393023bc3c4..731b9a43c610e7 100644 --- a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Basic.lean +++ b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Defs public import Mathlib.RingTheory.Spectrum.Maximal.Defs /-! - # Maximal ideal of local rings We prove basic properties of the maximal ideal of a local ring. diff --git a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Defs.lean b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Defs.lean index 43f39672d83532..5bbd869e96c28e 100644 --- a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Defs.lean +++ b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Defs.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.LocalRing.Basic /-! - # Maximal ideal of local rings We define the maximal ideal of a local ring as the ideal of all nonunits. diff --git a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Square.lean b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Square.lean index 0578c6c3037932..689c1d8be00bab 100644 --- a/Mathlib/RingTheory/LocalRing/MaximalIdeal/Square.lean +++ b/Mathlib/RingTheory/LocalRing/MaximalIdeal/Square.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic public import Mathlib.RingTheory.Nakayama /-! - # Lemmas about square of maximal ideal of local ring -/ diff --git a/Mathlib/RingTheory/LocalRing/NonLocalRing.lean b/Mathlib/RingTheory/LocalRing/NonLocalRing.lean index 6ea91aa7527f91..870a684b543128 100644 --- a/Mathlib/RingTheory/LocalRing/NonLocalRing.lean +++ b/Mathlib/RingTheory/LocalRing/NonLocalRing.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic public import Mathlib.RingTheory.Spectrum.Maximal.Basic /-! - # Non-local rings This file gathers some results about non-local rings. diff --git a/Mathlib/RingTheory/LocalRing/Quotient.lean b/Mathlib/RingTheory/LocalRing/Quotient.lean index 98fcbd0df701b9..50ad2b7be98ce7 100644 --- a/Mathlib/RingTheory/LocalRing/Quotient.lean +++ b/Mathlib/RingTheory/LocalRing/Quotient.lean @@ -17,7 +17,6 @@ public import Mathlib.RingTheory.Nakayama /-! - We gather results about the quotients of local rings. -/ diff --git a/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean b/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean index f0629b389ce617..ae80b067ca17b9 100644 --- a/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean +++ b/Mathlib/RingTheory/LocalRing/ResidueField/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.LocalRing.RingHom.Basic public import Mathlib.RingTheory.Ideal.Over /-! - # Residue Field of local rings We prove basic properties of the residue field of a local ring. diff --git a/Mathlib/RingTheory/LocalRing/ResidueField/Defs.lean b/Mathlib/RingTheory/LocalRing/ResidueField/Defs.lean index dd141c6c82a710..5f944fb7b98347 100644 --- a/Mathlib/RingTheory/LocalRing/ResidueField/Defs.lean +++ b/Mathlib/RingTheory/LocalRing/ResidueField/Defs.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic public import Mathlib.Tactic.CrossRefAttribute /-! - # Residue Field of local rings ## Main definitions diff --git a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean index deeaae7fdb0531..09da27d1b2b6b2 100644 --- a/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean +++ b/Mathlib/RingTheory/LocalRing/RingHom/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.LocalRing.MaximalIdeal.Basic public import Mathlib.RingTheory.Ideal.Maps /-! - # Local rings homomorphisms We prove basic properties of local rings homomorphisms. diff --git a/Mathlib/RingTheory/Localization/AsSubring.lean b/Mathlib/RingTheory/Localization/AsSubring.lean index 9cb5fec4087df7..570fcaeb0befe4 100644 --- a/Mathlib/RingTheory/Localization/AsSubring.lean +++ b/Mathlib/RingTheory/Localization/AsSubring.lean @@ -10,7 +10,6 @@ public import Mathlib.LinearAlgebra.FreeModule.Basic public import Mathlib.Algebra.Algebra.Subalgebra.Tower /-! - # Localizations of domains as subalgebras of the fraction field. Given a domain `A` with fraction field `K`, and a submonoid `S` of `A` which diff --git a/Mathlib/RingTheory/Localization/Finiteness.lean b/Mathlib/RingTheory/Localization/Finiteness.lean index 47bb4f0edf9e92..803537496ea081 100644 --- a/Mathlib/RingTheory/Localization/Finiteness.lean +++ b/Mathlib/RingTheory/Localization/Finiteness.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Localization.Algebra public import Mathlib.RingTheory.Localization.AtPrime.Basic /-! - # Finiteness properties under localization In this file we establish behaviour of `Module.Finite` under localizations. diff --git a/Mathlib/RingTheory/Localization/NormTrace.lean b/Mathlib/RingTheory/Localization/NormTrace.lean index 0ac9bdc4c3e206..c2cbcc4a12e580 100644 --- a/Mathlib/RingTheory/Localization/NormTrace.lean +++ b/Mathlib/RingTheory/Localization/NormTrace.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Norm.Basic public import Mathlib.RingTheory.Discriminant /-! - # Field/algebra norm / trace and localization This file contains results on the combination of `IsLocalization` and `Algebra.norm`, diff --git a/Mathlib/RingTheory/MvPolynomial/FreeCommRing.lean b/Mathlib/RingTheory/MvPolynomial/FreeCommRing.lean index 480af0ee207b81..4cda1c764b58c6 100644 --- a/Mathlib/RingTheory/MvPolynomial/FreeCommRing.lean +++ b/Mathlib/RingTheory/MvPolynomial/FreeCommRing.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.FreeCommRing /-! - # Constructing Ring terms from MvPolynomial This file provides tools for constructing ring terms that can be evaluated to particular diff --git a/Mathlib/RingTheory/MvPolynomial/Groebner.lean b/Mathlib/RingTheory/MvPolynomial/Groebner.lean index a6e9f9fd30e998..9407bd4594932b 100644 --- a/Mathlib/RingTheory/MvPolynomial/Groebner.lean +++ b/Mathlib/RingTheory/MvPolynomial/Groebner.lean @@ -41,7 +41,6 @@ The proof is done by induction, using two standard constructions * `MonomialOrder.div_single` is the variant of `MonomialOrder.div` for a single polynomial. - ## Reference : [Becker-Weispfenning1993] -/ diff --git a/Mathlib/RingTheory/MvPolynomial/Localization.lean b/Mathlib/RingTheory/MvPolynomial/Localization.lean index 77863f30d613c6..fd26db89b8e893 100644 --- a/Mathlib/RingTheory/MvPolynomial/Localization.lean +++ b/Mathlib/RingTheory/MvPolynomial/Localization.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Localization.BaseChange public import Mathlib.RingTheory.TensorProduct.MvPolynomial /-! - # Localization and multivariate polynomial rings In this file we show some results connecting multivariate polynomial rings and localization. diff --git a/Mathlib/RingTheory/MvPowerSeries/Basic.lean b/Mathlib/RingTheory/MvPowerSeries/Basic.lean index 08e0614e76f1b0..bbb136208677b0 100644 --- a/Mathlib/RingTheory/MvPowerSeries/Basic.lean +++ b/Mathlib/RingTheory/MvPowerSeries/Basic.lean @@ -48,7 +48,6 @@ We provide the natural inclusion from multivariate polynomials to multivariate f - `MvPolynomial.toMvPowerSeries`: the canonical coercion from `MvPolynomial` to `MvPowerSeries` - ## Note This file sets up the (semi)ring structure on multivariate power series: diff --git a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean index 92710c44b43b68..49fc25656f4d3b 100644 --- a/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean +++ b/Mathlib/RingTheory/MvPowerSeries/NoZeroDivisors.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.MvPowerSeries.Order a multivariate power series whose constant coefficient is not a zero divisor is itself not a zero divisor - - `MvPowerSeries.order_mul` : multiplicativity of `MvPowerSeries.order` if the semiring `R` has no zero divisors @@ -23,7 +22,6 @@ public import Mathlib.RingTheory.MvPowerSeries.Order If `R` has `NoZeroDivisors`, then so does `MvPowerSeries σ R`. - ## TODO * Transfer/adapt these results to `HahnSeries`. diff --git a/Mathlib/RingTheory/MvPowerSeries/Trunc.lean b/Mathlib/RingTheory/MvPowerSeries/Trunc.lean index 6da78eab0af087..427d0d14523bfb 100644 --- a/Mathlib/RingTheory/MvPowerSeries/Trunc.lean +++ b/Mathlib/RingTheory/MvPowerSeries/Trunc.lean @@ -13,7 +13,6 @@ public import Mathlib.Algebra.MvPolynomial.Degrees public import Mathlib.RingTheory.MvPowerSeries.Order /-! - # Formal (multivariate) power series - Truncation * `MvPowerSeries.truncFinset s p` restricts the support of a multivariate power series `p` @@ -38,7 +37,6 @@ public import Mathlib.RingTheory.MvPowerSeries.Order Here, `m` and `n` have types `σ →₀ ℕ` so that `m ≤ n` means that `m s ≤ n s` for all `s : σ`. - * `MvPowerSeries.coeff_mul_eq_coeff_trunc'_mul_trunc'` : compares the coefficients of a product with those of the product of truncations. diff --git a/Mathlib/RingTheory/Nilpotent/Basic.lean b/Mathlib/RingTheory/Nilpotent/Basic.lean index d373352c32d105..f5979fb661cd04 100644 --- a/Mathlib/RingTheory/Nilpotent/Basic.lean +++ b/Mathlib/RingTheory/Nilpotent/Basic.lean @@ -22,7 +22,6 @@ nilpotent elements are closed under many operations. For the definition of `nilradical`, see `Mathlib/RingTheory/Nilpotent/Lemmas.lean`. - ## Main definitions * `isNilpotent_neg_iff` diff --git a/Mathlib/RingTheory/Noetherian/Nilpotent.lean b/Mathlib/RingTheory/Noetherian/Nilpotent.lean index a42b49e232161d..9cefc17a02c1e6 100644 --- a/Mathlib/RingTheory/Noetherian/Nilpotent.lean +++ b/Mathlib/RingTheory/Noetherian/Nilpotent.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Noetherian.Defs /-! # Nilpotent ideals in Noetherian rings - ## Main results * `IsNoetherianRing.isNilpotent_nilradical` diff --git a/Mathlib/RingTheory/OreLocalization/Basic.lean b/Mathlib/RingTheory/OreLocalization/Basic.lean index e966bb80606740..3adbd8e3dc9366 100644 --- a/Mathlib/RingTheory/OreLocalization/Basic.lean +++ b/Mathlib/RingTheory/OreLocalization/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.OreLocalization.Basic public import Mathlib.Algebra.GroupWithZero.Defs /-! - # Localization over left Ore sets. This file proves results on the localization of rings (monoids with zeros) over a left Ore set. @@ -20,7 +19,6 @@ This file proves results on the localization of rings (monoids with zeros) over * * [Zoran Škoda, *Noncommutative localization in noncommutative geometry*][skoda2006] - ## Tags localization, Ore, non-commutative diff --git a/Mathlib/RingTheory/OreLocalization/OreSet.lean b/Mathlib/RingTheory/OreLocalization/OreSet.lean index ab74b4f90def7a..69721a64a519f6 100644 --- a/Mathlib/RingTheory/OreLocalization/OreSet.lean +++ b/Mathlib/RingTheory/OreLocalization/OreSet.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Ring.Regular public import Mathlib.GroupTheory.OreLocalization.OreSet /-! - # (Left) Ore sets and rings This file contains results on left Ore sets for rings and monoids with zero. diff --git a/Mathlib/RingTheory/OreLocalization/Ring.lean b/Mathlib/RingTheory/OreLocalization/Ring.lean index b0baae937aea9f..10b83e52ab01ba 100644 --- a/Mathlib/RingTheory/OreLocalization/Ring.lean +++ b/Mathlib/RingTheory/OreLocalization/Ring.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Field.Defs public import Mathlib.RingTheory.OreLocalization.NonZeroDivisors /-! - # Module and Ring instances of Ore Localizations The `Monoid` and `DistribMulAction` instances and additive versions are provided in diff --git a/Mathlib/RingTheory/OrzechProperty.lean b/Mathlib/RingTheory/OrzechProperty.lean index 8e6baa2b9e669a..ba89e2f2509a0f 100644 --- a/Mathlib/RingTheory/OrzechProperty.lean +++ b/Mathlib/RingTheory/OrzechProperty.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Module.TransferInstance public import Mathlib.RingTheory.Finiteness.Cardinality /-! - # Orzech property of rings In this file we define the following property of rings: diff --git a/Mathlib/RingTheory/Perfectoid/BDeRham.lean b/Mathlib/RingTheory/Perfectoid/BDeRham.lean index abd5ee140684bf..3f4bda0834c209 100644 --- a/Mathlib/RingTheory/Perfectoid/BDeRham.lean +++ b/Mathlib/RingTheory/Perfectoid/BDeRham.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Localization.Away.Basic public import Mathlib.RingTheory.Perfectoid.FontaineTheta /-! - # The de Rham Period Ring $\mathbb{B}_{dR}^+$ and $\mathbb{B}_{dR}$ In this file, we define the de Rham period ring $\mathbb{B}_{dR}^+$ and diff --git a/Mathlib/RingTheory/Polynomial/Dickson.lean b/Mathlib/RingTheory/Polynomial/Dickson.lean index 99c6d0c632f322..85bf7b576acc24 100644 --- a/Mathlib/RingTheory/Polynomial/Dickson.lean +++ b/Mathlib/RingTheory/Polynomial/Dickson.lean @@ -109,7 +109,6 @@ theorem dickson_two_zero : ∀ n : ℕ, dickson 2 (0 : R) n = X ^ n section Dickson /-! - ### A Lambda structure on `ℤ[X]` Mathlib doesn't currently know what a Lambda ring is. diff --git a/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.lean b/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.lean index 1afbd10b315879..9a0b4446fc52b6 100644 --- a/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.lean +++ b/Mathlib/RingTheory/Polynomial/Eisenstein/Distinguished.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.Polynomial.Eisenstein.Basic public import Mathlib.RingTheory.PowerSeries.Order /-! - # Distinguished polynomial In this file we define the predicate `Polynomial.IsDistinguishedAt` diff --git a/Mathlib/RingTheory/Polynomial/IrreducibleRing.lean b/Mathlib/RingTheory/Polynomial/IrreducibleRing.lean index 65a710308d87bf..727a6f0d50e1cf 100644 --- a/Mathlib/RingTheory/Polynomial/IrreducibleRing.lean +++ b/Mathlib/RingTheory/Polynomial/IrreducibleRing.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Polynomial.Eval.Irreducible public import Mathlib.RingTheory.Polynomial.Nilpotent /-! - # Polynomials over an irreducible ring This file contains results about the polynomials over an irreducible ring (i.e. a ring with only diff --git a/Mathlib/RingTheory/Polynomial/IsIntegral.lean b/Mathlib/RingTheory/Polynomial/IsIntegral.lean index 1bda775fc4645b..7229081b45728c 100644 --- a/Mathlib/RingTheory/Polynomial/IsIntegral.lean +++ b/Mathlib/RingTheory/Polynomial/IsIntegral.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Polynomial.RationalRoot public import Mathlib.RingTheory.IntegralClosure.IsIntegral.AlmostIntegral /-! - # Results about coefficients of polynomials being integral ## Main results diff --git a/Mathlib/RingTheory/Polynomial/SeparableDegree.lean b/Mathlib/RingTheory/Polynomial/SeparableDegree.lean index 85c755c07fd0dc..eb5676dea4f445 100644 --- a/Mathlib/RingTheory/Polynomial/SeparableDegree.lean +++ b/Mathlib/RingTheory/Polynomial/SeparableDegree.lean @@ -9,7 +9,6 @@ public import Mathlib.Algebra.Algebra.Defs public import Mathlib.FieldTheory.Separable /-! - # Separable degree This file contains basics about the separable degree of a polynomial. diff --git a/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean b/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean index cc868eaae619d9..840e2fd3a49377 100644 --- a/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean +++ b/Mathlib/RingTheory/Polynomial/UniversalFactorizationRing.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.LocalRing.ResidueField.Ideal /-! - # Universal factorization ring Let `R` be a commutative ring and `p : R[X]` be monic of degree `n` and let `n = m + k`. diff --git a/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean b/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean index cff3f91d0f39c4..d2f8852c615531 100644 --- a/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean +++ b/Mathlib/RingTheory/PowerSeries/CoeffMulMem.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.Ideal.BigOperators public import Mathlib.RingTheory.PowerSeries.Basic /-! - # Some results on the coefficients of multiplication of two power series ## Main results diff --git a/Mathlib/RingTheory/PowerSeries/Ideal.lean b/Mathlib/RingTheory/PowerSeries/Ideal.lean index a14c067a3221b7..48edb2440892c3 100644 --- a/Mathlib/RingTheory/PowerSeries/Ideal.lean +++ b/Mathlib/RingTheory/PowerSeries/Ideal.lean @@ -21,7 +21,6 @@ generated by `n` elements if `X ∉ I` and `n + 1` elements otherwise). This imp `R⟦X⟧` is noetherian if `R` is. The proof of `eq_of_le_of_X_notMem_of_fg_of_isPrime` is inspired by [zbMATH05171613]. - ## Main results * `PowerSeries.eq_span_insert_X_of_X_mem_of_span_eq` : given an ideal `I` of `R⟦X⟧` such that @@ -33,7 +32,6 @@ generated by `n` elements if `X ∉ I` and `n + 1` elements otherwise). This imp * `PowerSeries.fg_iff_of_isPrime` : a prime ideal `P` of `R⟦X⟧` is finitely generated if and only if `P.map constantCoeff` is finitely generated. - ## Implementation details There is a simpler proof of `IsNoetherianRing R → IsNoetherianRing R⟦X⟧`, similar to that for diff --git a/Mathlib/RingTheory/PowerSeries/Trunc.lean b/Mathlib/RingTheory/PowerSeries/Trunc.lean index f8bbbd3f6c470a..340537d2ba1e58 100644 --- a/Mathlib/RingTheory/PowerSeries/Trunc.lean +++ b/Mathlib/RingTheory/PowerSeries/Trunc.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Polynomial.Degree.Lemmas public import Mathlib.RingTheory.PowerSeries.Basic /-! - # Formal power series in one variable - Truncation `PowerSeries.trunc n φ` truncates a (univariate) formal power series diff --git a/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean b/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean index ac4b409fb6d2a3..d3672c9b7894cf 100644 --- a/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean +++ b/Mathlib/RingTheory/PowerSeries/WeierstrassPreparation.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.PowerSeries.Inverse public import Mathlib.RingTheory.PowerSeries.Trunc /-! - # Weierstrass preparation theorem for power series over a complete local ring In this file we define Weierstrass division, Weierstrass factorization, and prove @@ -98,7 +97,6 @@ namespace PowerSeries variable {A : Type*} [CommRing A] /-! - ## Weierstrass division -/ @@ -639,7 +637,6 @@ alias zero_weierstrassMod := weierstrassMod_zero_left end IsLocalRing /-! - ## Weierstrass preparation theorem -/ diff --git a/Mathlib/RingTheory/QuasiFinite/Weakly.lean b/Mathlib/RingTheory/QuasiFinite/Weakly.lean index 899aaacb703f7a..b2a769ba2e967d 100644 --- a/Mathlib/RingTheory/QuasiFinite/Weakly.lean +++ b/Mathlib/RingTheory/QuasiFinite/Weakly.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.QuasiFinite.Basic /-! - # Weakly Quasi-finite primes The definition `Algebra.QuasiFiniteAt` is equivalent to the usual definition "isolated in fibers" diff --git a/Mathlib/RingTheory/ReesAlgebra.lean b/Mathlib/RingTheory/ReesAlgebra.lean index 148cfd9c7978ca..87adcfec868509 100644 --- a/Mathlib/RingTheory/ReesAlgebra.lean +++ b/Mathlib/RingTheory/ReesAlgebra.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Ideal.BigOperators public import Mathlib.RingTheory.FiniteType /-! - # Rees algebra The Rees algebra of an ideal `I` is the subalgebra `R[It]` of `R[t]` defined as `R[It] = ⨁ₙ Iⁿ tⁿ`. diff --git a/Mathlib/RingTheory/Regular/LinearMap.lean b/Mathlib/RingTheory/Regular/LinearMap.lean index 9835b42022974a..0f1e89c92ec462 100644 --- a/Mathlib/RingTheory/Regular/LinearMap.lean +++ b/Mathlib/RingTheory/Regular/LinearMap.lean @@ -14,7 +14,6 @@ public import Mathlib.RingTheory.Regular.IsSMulRegular public import Mathlib.RingTheory.Support /-! - # Hom(N,M) is subsingleton iff there exists a smul regular element of M in ann(N) Let `M` and `N` be `R`-modules. In this section we prove that `Hom(N,M)` is subsingleton iff diff --git a/Mathlib/RingTheory/Regular/ProjectiveDimension.lean b/Mathlib/RingTheory/Regular/ProjectiveDimension.lean index eb45a09437398d..83092a5e5f2419 100644 --- a/Mathlib/RingTheory/Regular/ProjectiveDimension.lean +++ b/Mathlib/RingTheory/Regular/ProjectiveDimension.lean @@ -14,7 +14,6 @@ public import Mathlib.RingTheory.Regular.Category public import Mathlib.RingTheory.Regular.RegularSequence /-! - # ProjectiveDimension of quotient by regular element For `M` a finitely generated module over Noetherian local ring `R` and an `M`-regular element `x` diff --git a/Mathlib/RingTheory/RingHom/Finite.lean b/Mathlib/RingTheory/RingHom/Finite.lean index 0f28149a7f48b5..209ccc9e9e3006 100644 --- a/Mathlib/RingTheory/RingHom/Finite.lean +++ b/Mathlib/RingTheory/RingHom/Finite.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.LocalProperties.Basic public import Mathlib.RingTheory.TensorProduct.Finite /-! - # The meta properties of finite ring homomorphisms. ## Main results diff --git a/Mathlib/RingTheory/RingHom/FinitePresentation.lean b/Mathlib/RingTheory/RingHom/FinitePresentation.lean index a2e851d1f3d944..cc6a8a440a5dd6 100644 --- a/Mathlib/RingTheory/RingHom/FinitePresentation.lean +++ b/Mathlib/RingTheory/RingHom/FinitePresentation.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Localization.Away.AdjoinRoot public import Mathlib.RingTheory.Finiteness.FinitePresentationLocal /-! - # The meta properties of finitely-presented ring homomorphisms. The main result is `RingHom.finitePresentation_isLocal`. diff --git a/Mathlib/RingTheory/RingHom/FiniteType.lean b/Mathlib/RingTheory/RingHom/FiniteType.lean index 2b94ddb5958f73..57dad1548879fe 100644 --- a/Mathlib/RingTheory/RingHom/FiniteType.lean +++ b/Mathlib/RingTheory/RingHom/FiniteType.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Localization.InvSubmonoid public import Mathlib.RingTheory.RingHom.Finite /-! - # The meta properties of finite-type ring homomorphisms. ## Main results diff --git a/Mathlib/RingTheory/RingHom/Integral.lean b/Mathlib/RingTheory/RingHom/Integral.lean index 2ff7ff779b31bd..4f4574681a6143 100644 --- a/Mathlib/RingTheory/RingHom/Integral.lean +++ b/Mathlib/RingTheory/RingHom/Integral.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.LocalProperties.Basic public import Mathlib.RingTheory.Localization.Integral /-! - # The meta properties of integral ring homomorphisms. -/ diff --git a/Mathlib/RingTheory/RingHom/Surjective.lean b/Mathlib/RingTheory/RingHom/Surjective.lean index 96a4740065392f..9d82c64e99962b 100644 --- a/Mathlib/RingTheory/RingHom/Surjective.lean +++ b/Mathlib/RingTheory/RingHom/Surjective.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.LocalProperties.Basic /-! - # The meta properties of surjective ring homomorphisms. ## Main results diff --git a/Mathlib/RingTheory/RingHom/Unramified.lean b/Mathlib/RingTheory/RingHom/Unramified.lean index bd73f63ea5d797..8b2a54210d7023 100644 --- a/Mathlib/RingTheory/RingHom/Unramified.lean +++ b/Mathlib/RingTheory/RingHom/Unramified.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Unramified.Locus public import Mathlib.RingTheory.LocalProperties.Basic /-! - # The meta properties of unramified ring homomorphisms. -/ diff --git a/Mathlib/RingTheory/RootsOfUnity/CyclotomicUnits.lean b/Mathlib/RingTheory/RootsOfUnity/CyclotomicUnits.lean index 2b2c98494c996d..c6a37d9f4b378c 100644 --- a/Mathlib/RingTheory/RootsOfUnity/CyclotomicUnits.lean +++ b/Mathlib/RingTheory/RootsOfUnity/CyclotomicUnits.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots We gather miscellaneous results about units given by sums of powers of roots of unit, the so-called *cyclotomic units*. - ## Main results * `IsPrimitiveRoot.associated_sub_one_pow_sub_one_of_coprime` : given an `n`-th primitive root of diff --git a/Mathlib/RingTheory/Smooth/Basic.lean b/Mathlib/RingTheory/Smooth/Basic.lean index 54640dfe993ab0..cfd0475ee3ec14 100644 --- a/Mathlib/RingTheory/Smooth/Basic.lean +++ b/Mathlib/RingTheory/Smooth/Basic.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Smooth.Kaehler public import Mathlib.RingTheory.Unramified.Basic /-! - # Smooth morphisms An `R`-algebra `A` is formally smooth if `Ω[A⁄R]` is `A`-projective and `H¹(L_{A/R}) = 0`. diff --git a/Mathlib/RingTheory/Smooth/Fiber.lean b/Mathlib/RingTheory/Smooth/Fiber.lean index 0fb987b25ad092..2774c6979ca0c9 100644 --- a/Mathlib/RingTheory/Smooth/Fiber.lean +++ b/Mathlib/RingTheory/Smooth/Fiber.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Smooth.Local public import Mathlib.RingTheory.Etale.Locus /-! - # Flat and smooth fibers imply smooth ## Main results diff --git a/Mathlib/RingTheory/Smooth/Field.lean b/Mathlib/RingTheory/Smooth/Field.lean index ca760870af23e6..4dbe0b8cbe8543 100644 --- a/Mathlib/RingTheory/Smooth/Field.lean +++ b/Mathlib/RingTheory/Smooth/Field.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Etale.Field public import Mathlib.FieldTheory.SeparablyGenerated /-! - # Smooth algebras over fields We show that separably generated extensions of fields are smooth. diff --git a/Mathlib/RingTheory/Smooth/Flat.lean b/Mathlib/RingTheory/Smooth/Flat.lean index df4ed4f7919b2a..65db32efdbfedc 100644 --- a/Mathlib/RingTheory/Smooth/Flat.lean +++ b/Mathlib/RingTheory/Smooth/Flat.lean @@ -26,7 +26,6 @@ The proof proceeds in two steps: 2. In the general case, we choose a model of `A` over a finitely generated `ℤ`-subalgebra of `R` and apply 1. - ## References - [Conde-Lago, A short proof of smooth implies flat][condelago2016shortproofsmoothimplies] diff --git a/Mathlib/RingTheory/Smooth/Kaehler.lean b/Mathlib/RingTheory/Smooth/Kaehler.lean index a183f5d5c81e51..dc5dbb94aca595 100644 --- a/Mathlib/RingTheory/Smooth/Kaehler.lean +++ b/Mathlib/RingTheory/Smooth/Kaehler.lean @@ -31,7 +31,6 @@ public import Mathlib.RingTheory.Extension.Cotangent.Basic - https://stacks.math.columbia.edu/tag/00TH - [B. Iversen, *Generic Local Structure of the Morphisms in Commutative Algebra*][iversen] - -/ @[expose] public section diff --git a/Mathlib/RingTheory/Smooth/Pi.lean b/Mathlib/RingTheory/Smooth/Pi.lean index f4799e26092736..098e69fcc408f9 100644 --- a/Mathlib/RingTheory/Smooth/Pi.lean +++ b/Mathlib/RingTheory/Smooth/Pi.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Idempotents public import Mathlib.RingTheory.Smooth.Basic /-! - # Formal-smoothness of finite products of rings ## Main result diff --git a/Mathlib/RingTheory/Smooth/Quotient.lean b/Mathlib/RingTheory/Smooth/Quotient.lean index fc847cd5f77427..392ade3b8d915f 100644 --- a/Mathlib/RingTheory/Smooth/Quotient.lean +++ b/Mathlib/RingTheory/Smooth/Quotient.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.RingHom.Flat public import Mathlib.RingTheory.RingHom.Smooth /-! - # Some lemmas about formally smooth under quotient In this file, we formalize the result [Stacks 031L] : For flat ring homomorphism `f : R →+* S`, diff --git a/Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean b/Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean index 0bd9b1d81f856c..80815cd3f77d77 100644 --- a/Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean +++ b/Mathlib/RingTheory/Spectrum/Prime/FreeLocus.lean @@ -18,7 +18,6 @@ public import Mathlib.RingTheory.TensorProduct.IsBaseChangePi public import Mathlib.RingTheory.Support /-! - # The free locus of a module ## Main definitions and results diff --git a/Mathlib/RingTheory/Spectrum/Prime/Module.lean b/Mathlib/RingTheory/Spectrum/Prime/Module.lean index 830b1a4a766eb4..b6a4725f2a10b5 100644 --- a/Mathlib/RingTheory/Spectrum/Prime/Module.lean +++ b/Mathlib/RingTheory/Spectrum/Prime/Module.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.Topology public import Mathlib.RingTheory.Support /-! - # Subsets of prime spectra related to modules ## Main results diff --git a/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean b/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean index d8ec4669009cfa..0e714796558382 100644 --- a/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean +++ b/Mathlib/RingTheory/Spectrum/Prime/Polynomial.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.Topology public import Mathlib.RingTheory.TensorProduct.MvPolynomial /-! - # Prime spectrum of (multivariate) polynomials Also see `AlgebraicGeometry/AffineSpace` for the affine space over arbitrary schemes. diff --git a/Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean b/Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean index 1b9ba5cfd62f36..d8e74cf5b17753 100644 --- a/Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean +++ b/Mathlib/RingTheory/Spectrum/Prime/TensorProduct.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.Topology public import Mathlib.RingTheory.SurjectiveOnStalks /-! - # Lemmas regarding the prime spectrum of tensor products ## Main result diff --git a/Mathlib/RingTheory/Support.lean b/Mathlib/RingTheory/Support.lean index b5e49ab6706dab..123b0f7d02cdc7 100644 --- a/Mathlib/RingTheory/Support.lean +++ b/Mathlib/RingTheory/Support.lean @@ -14,7 +14,6 @@ public import Mathlib.RingTheory.Spectrum.Prime.Basic public import Mathlib.RingTheory.LocalProperties.Basic /-! - # Support of a module ## Main results diff --git a/Mathlib/RingTheory/TensorProduct/IsBaseChangeRightExact.lean b/Mathlib/RingTheory/TensorProduct/IsBaseChangeRightExact.lean index acee995f971245..2a7107125a00f9 100644 --- a/Mathlib/RingTheory/TensorProduct/IsBaseChangeRightExact.lean +++ b/Mathlib/RingTheory/TensorProduct/IsBaseChangeRightExact.lean @@ -11,7 +11,6 @@ public import Mathlib.LinearAlgebra.TensorProduct.RightExactness public import Mathlib.RingTheory.IsTensorProduct /-! - # Lemmas about `IsBaseChange` under exact sequences In this file, we show that for an `R`-algebra `S` taking cokernels commutes with base change diff --git a/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean b/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean index cf25c5461a1c52..d91b138852ed46 100644 --- a/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean +++ b/Mathlib/RingTheory/TensorProduct/MvPolynomial.lean @@ -12,7 +12,6 @@ public import Mathlib.Algebra.MvPolynomial.Equiv public import Mathlib.RingTheory.IsTensorProduct /-! - # Tensor Product of (multivariate) polynomial rings Let `Semiring R`, `Algebra R S` and `Module R N`. diff --git a/Mathlib/RingTheory/TensorProduct/Nontrivial.lean b/Mathlib/RingTheory/TensorProduct/Nontrivial.lean index a32958caefdce0..643c3f8fa0797b 100644 --- a/Mathlib/RingTheory/TensorProduct/Nontrivial.lean +++ b/Mathlib/RingTheory/TensorProduct/Nontrivial.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.Localization.FractionRing public import Mathlib.Algebra.Ring.Hom.InjSurj /-! - # Nontriviality of tensor product of algebras This file contains some more results on nontriviality of tensor product of algebras. diff --git a/Mathlib/RingTheory/Trace/Quotient.lean b/Mathlib/RingTheory/Trace/Quotient.lean index b54e87e655bc4a..4687a4b031f1e2 100644 --- a/Mathlib/RingTheory/Trace/Quotient.lean +++ b/Mathlib/RingTheory/Trace/Quotient.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.LocalRing.Quotient public import Mathlib.RingTheory.Localization.AtPrime.Basic /-! - We gather results about the relations between the trace map on `B → A` and the trace map on quotients and localizations. diff --git a/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean b/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean index b9eca70f6fb1fa..688e67a494ecec 100644 --- a/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean +++ b/Mathlib/RingTheory/TwoSidedIdeal/Operations.lean @@ -29,7 +29,6 @@ This file defines operations on two-sided ideals of a ring `R`. element `x` is in the two-sided ideal spanned by `s` if and only if `x` is in the closure of `{a*y*b | a, b ∈ R, y ∈ s}` as an additive subgroup. - - `TwoSidedIdeal.comap`: pullback of a two-sided ideal; defined as the preimage of a two-sided ideal. - `TwoSidedIdeal.map`: pushforward of a two-sided ideal; defined as the span of the image of a diff --git a/Mathlib/RingTheory/UniqueFactorizationDomain/Kaplansky.lean b/Mathlib/RingTheory/UniqueFactorizationDomain/Kaplansky.lean index 84c48a1f3838af..560d0635e46d05 100644 --- a/Mathlib/RingTheory/UniqueFactorizationDomain/Kaplansky.lean +++ b/Mathlib/RingTheory/UniqueFactorizationDomain/Kaplansky.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.UniqueFactorizationDomain.Basic public import Mathlib.RingTheory.UniqueFactorizationDomain.Ideal /-! - # Kaplansky criterion for factoriality We prove Kaplansky criterion for factoriality: an integral domain is a UFD if and only if every diff --git a/Mathlib/RingTheory/Unramified/Basic.lean b/Mathlib/RingTheory/Unramified/Basic.lean index e92075346fa9e3..4d6c182ea79c8e 100644 --- a/Mathlib/RingTheory/Unramified/Basic.lean +++ b/Mathlib/RingTheory/Unramified/Basic.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.TensorProduct.Quotient public import Mathlib.Algebra.Algebra.Shrink /-! - # Unramified morphisms An `R`-algebra `A` is formally unramified if `Ω[A⁄R]` is trivial. diff --git a/Mathlib/RingTheory/Unramified/LocalStructure.lean b/Mathlib/RingTheory/Unramified/LocalStructure.lean index 32967415cc43c2..781d7c72eb5c91 100644 --- a/Mathlib/RingTheory/Unramified/LocalStructure.lean +++ b/Mathlib/RingTheory/Unramified/LocalStructure.lean @@ -12,7 +12,6 @@ public import Mathlib.RingTheory.Unramified.LocalRing public import Mathlib.RingTheory.ZariskisMainTheorem /-! - # Local structure of unramified algebras In this file, we will prove that if `S` is a finite type `R`-algebra unramified at `Q`, then diff --git a/Mathlib/RingTheory/Unramified/Pi.lean b/Mathlib/RingTheory/Unramified/Pi.lean index 0dcaf171bbf480..b21c93d2ff3d52 100644 --- a/Mathlib/RingTheory/Unramified/Pi.lean +++ b/Mathlib/RingTheory/Unramified/Pi.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.Unramified.Basic /-! - # Formal-unramification of finite products of rings ## Main result diff --git a/Mathlib/RingTheory/Valuation/Basic.lean b/Mathlib/RingTheory/Valuation/Basic.lean index a669fb8572cbfd..bb91975c43e26c 100644 --- a/Mathlib/RingTheory/Valuation/Basic.lean +++ b/Mathlib/RingTheory/Valuation/Basic.lean @@ -13,7 +13,6 @@ public import Mathlib.RingTheory.Ideal.Maps public import Mathlib.Tactic.TFAE /-! - # The basics of valuation theory. The basic theory of valuations (non-archimedean norms) on a commutative ring, diff --git a/Mathlib/RingTheory/Valuation/Discrete/Basic.lean b/Mathlib/RingTheory/Valuation/Discrete/Basic.lean index dc93e62bd2767d..55b0a2b157fb74 100644 --- a/Mathlib/RingTheory/Valuation/Discrete/Basic.lean +++ b/Mathlib/RingTheory/Valuation/Discrete/Basic.lean @@ -51,7 +51,6 @@ discrete, we use the name `IsRankOneDiscrete` to refer to discrete valuations in valuation ring. This instance is the formalization of Chapter I, Section 1, Proposition 1 in [serre1968]. - ## TODO * Relate discrete valuations and discrete valuation rings (contained in the project ) diff --git a/Mathlib/RingTheory/Valuation/DiscreteValuativeRel.lean b/Mathlib/RingTheory/Valuation/DiscreteValuativeRel.lean index 319f86006fccf6..425c2d8821aaf1 100644 --- a/Mathlib/RingTheory/Valuation/DiscreteValuativeRel.lean +++ b/Mathlib/RingTheory/Valuation/DiscreteValuativeRel.lean @@ -10,7 +10,6 @@ public import Mathlib.GroupTheory.ArchimedeanDensely public import Mathlib.RingTheory.Valuation.RankOne /-! - # Discrete Valuative Relations Discrete valuative relations have a maximal element less than one in the value group. diff --git a/Mathlib/RingTheory/Valuation/ExtendToLocalization.lean b/Mathlib/RingTheory/Valuation/ExtendToLocalization.lean index def1529b422cb4..1afb44fa584107 100644 --- a/Mathlib/RingTheory/Valuation/ExtendToLocalization.lean +++ b/Mathlib/RingTheory/Valuation/ExtendToLocalization.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.Localization.Defs public import Mathlib.RingTheory.Valuation.Basic /-! - # Extending valuations to a localization We show that, given a valuation `v` taking values in a linearly ordered commutative *group* diff --git a/Mathlib/RingTheory/Valuation/LocalSubring.lean b/Mathlib/RingTheory/Valuation/LocalSubring.lean index 2b3bd4613aba31..2824ff05e8347f 100644 --- a/Mathlib/RingTheory/Valuation/LocalSubring.lean +++ b/Mathlib/RingTheory/Valuation/LocalSubring.lean @@ -18,7 +18,6 @@ public import Mathlib.RingTheory.Valuation.ValuationSubring set_option linter.style.header false /-! - # Valuation subrings are exactly the maximal local subrings See `LocalSubring.isMax_iff`. diff --git a/Mathlib/RingTheory/Valuation/ValuationSubring.lean b/Mathlib/RingTheory/Valuation/ValuationSubring.lean index cf0ce5bbf3e112..31228d8d1e5b46 100644 --- a/Mathlib/RingTheory/Valuation/ValuationSubring.lean +++ b/Mathlib/RingTheory/Valuation/ValuationSubring.lean @@ -15,7 +15,6 @@ public import Mathlib.RingTheory.KrullDimension.Basic public import Mathlib.RingTheory.Spectrum.Prime.Topology /-! - # Valuation subrings of a field ## Projects diff --git a/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean b/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean index 0cb1d9d4957031..3d3369a799738c 100644 --- a/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean +++ b/Mathlib/RingTheory/Valuation/ValuativeRel/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Defs.Filter public import Mathlib.Order.Filter.Bases.Basic /-! - # Valuative Relations In this file we introduce a class called `ValuativeRel R` for a ring `R`. @@ -37,7 +36,6 @@ saying that `R` is endowed with an equivalence class of valuations. `[ValuativeExtension A B]` ensures that the algebra map `A → B` is compatible with the valuations on `A` and `B`. For example, this can be used to talk about extensions of valued fields. - ## Remark The last two axioms in `ValuativeRel`, namely `vle_mul_cancel` and `not_vle_one_zero`, are diff --git a/Mathlib/RingTheory/Valuation/ValuativeRel/Trivial.lean b/Mathlib/RingTheory/Valuation/ValuativeRel/Trivial.lean index 35e6c1e2c7a42e..a48ce0232e66d5 100644 --- a/Mathlib/RingTheory/Valuation/ValuativeRel/Trivial.lean +++ b/Mathlib/RingTheory/Valuation/ValuativeRel/Trivial.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.Valuation.ValuativeRel.Basic /-! - # Trivial Valuative Relations Trivial valuative relations relate all non-zero elements to each other. Equivalently, diff --git a/Mathlib/RingTheory/WittVector/Compare.lean b/Mathlib/RingTheory/WittVector/Compare.lean index f5084263c587bf..164d4cb73c4903 100644 --- a/Mathlib/RingTheory/WittVector/Compare.lean +++ b/Mathlib/RingTheory/WittVector/Compare.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.WittVector.Identities public import Mathlib.NumberTheory.Padics.RingHoms /-! - # Comparison isomorphism between `WittVector p (ZMod p)` and `ℤ_[p]` We construct a ring isomorphism between `WittVector p (ZMod p)` and `ℤ_[p]`. diff --git a/Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean b/Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean index 8ea43e8cd2eebd..f3e2eea9fe6de7 100644 --- a/Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean +++ b/Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean @@ -11,7 +11,6 @@ public import Mathlib.RingTheory.DiscreteValuationRing.Basic public import Mathlib.Tactic.LinearCombination /-! - # Witt vectors over a perfect ring This file establishes that Witt vectors over a perfect field are a discrete valuation ring. diff --git a/Mathlib/RingTheory/WittVector/Domain.lean b/Mathlib/RingTheory/WittVector/Domain.lean index 1596872f2caa2b..0acb7a3c646863 100644 --- a/Mathlib/RingTheory/WittVector/Domain.lean +++ b/Mathlib/RingTheory/WittVector/Domain.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.WittVector.Identities /-! - # Witt vectors over a domain This file builds to the proof `WittVector.instIsDomain`, diff --git a/Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean b/Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean index 25380ba5a64305..b795372581b54e 100644 --- a/Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean +++ b/Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean @@ -51,7 +51,6 @@ local notation "𝕎" => WittVector p namespace RecursionMain /-! - ## The recursive case of the vector coefficients The first coefficient of our solution vector is easy to define below. diff --git a/Mathlib/RingTheory/WittVector/InitTail.lean b/Mathlib/RingTheory/WittVector/InitTail.lean index c3117a433dd17d..ece9753f66be35 100644 --- a/Mathlib/RingTheory/WittVector/InitTail.lean +++ b/Mathlib/RingTheory/WittVector/InitTail.lean @@ -9,7 +9,6 @@ public import Mathlib.RingTheory.WittVector.Basic public import Mathlib.RingTheory.WittVector.IsPoly /-! - # `init` and `tail` Given a Witt vector `x`, we are sometimes interested diff --git a/Mathlib/RingTheory/WittVector/Isocrystal.lean b/Mathlib/RingTheory/WittVector/Isocrystal.lean index a91b60b777cdcc..5cdbad76cb820d 100644 --- a/Mathlib/RingTheory/WittVector/Isocrystal.lean +++ b/Mathlib/RingTheory/WittVector/Isocrystal.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.WittVector.FrobeniusFractionField /-! - ## F-isocrystals over a perfect field When `k` is an integral domain, so is `𝕎 k`, and we can consider its field of fractions `K(p, k)`. diff --git a/Mathlib/RingTheory/WittVector/Truncated.lean b/Mathlib/RingTheory/WittVector/Truncated.lean index c5bf274255efd8..4c8ac71df63545 100644 --- a/Mathlib/RingTheory/WittVector/Truncated.lean +++ b/Mathlib/RingTheory/WittVector/Truncated.lean @@ -8,7 +8,6 @@ module public import Mathlib.RingTheory.WittVector.InitTail /-! - # Truncated Witt vectors The ring of truncated Witt vectors (of length `n`) is a quotient of the ring of Witt vectors. diff --git a/Mathlib/RingTheory/WittVector/WittPolynomial.lean b/Mathlib/RingTheory/WittVector/WittPolynomial.lean index 9c2d845b0c49f1..c70ce26f9382ba 100644 --- a/Mathlib/RingTheory/WittVector/WittPolynomial.lean +++ b/Mathlib/RingTheory/WittVector/WittPolynomial.lean @@ -180,7 +180,6 @@ end PPrime end /-! - ## Witt polynomials as a basis of the polynomial algebra If `p` is invertible in `R`, then the Witt polynomials form a basis diff --git a/Mathlib/Tactic/Algebraize.lean b/Mathlib/Tactic/Algebraize.lean index 54b80d0fbf5734..3619033c7fa3ec 100644 --- a/Mathlib/Tactic/Algebraize.lean +++ b/Mathlib/Tactic/Algebraize.lean @@ -10,7 +10,6 @@ public meta import Mathlib.Tactic.Attr.Core public meta import Mathlib.Tactic.ToAdditive /-! - ## Algebraize tactic This file defines the `algebraize` tactic. The basic functionality of this tactic is to diff --git a/Mathlib/Tactic/ApplyFun.lean b/Mathlib/Tactic/ApplyFun.lean index b06d1fcc9a7e6b..108f933456c7c1 100644 --- a/Mathlib/Tactic/ApplyFun.lean +++ b/Mathlib/Tactic/ApplyFun.lean @@ -199,7 +199,6 @@ Apply a function to an equality or inequality in either a local hypothesis or th `apply_fun f` will replace the goal with `f a ≤ f b`. If `f` is anything else (e.g. just a function, or an `Equiv`), `apply_fun` will fail. - Typical usage is: ```lean open Function diff --git a/Mathlib/Tactic/Change.lean b/Mathlib/Tactic/Change.lean index f00a3581dd86ce..f7d74e2fdcd614 100644 --- a/Mathlib/Tactic/Change.lean +++ b/Mathlib/Tactic/Change.lean @@ -9,7 +9,6 @@ public import Mathlib.Init public meta import Lean.Elab.Tactic.ElabTerm public meta import Lean.Meta.Tactic.TryThis /-! - # Tactic `change? term` This tactic is used to suggest a replacement of the goal by a definitionally equal term. diff --git a/Mathlib/Tactic/ClickSuggestions/Unfold.lean b/Mathlib/Tactic/ClickSuggestions/Unfold.lean index 91c4a0a3e7f843..37081c04633cb9 100644 --- a/Mathlib/Tactic/ClickSuggestions/Unfold.lean +++ b/Mathlib/Tactic/ClickSuggestions/Unfold.lean @@ -16,7 +16,6 @@ public import ProofWidgets.Component.OfRpcMethod public import Mathlib.Tactic.ClickSuggestions.Util /-! - # Interactive unfolding This file defines the interactive tactic `unfold?`. diff --git a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Defs.lean b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Defs.lean index 2264a94ec0a599..ebd416cd6d985e 100644 --- a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Defs.lean +++ b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Defs.lean @@ -9,7 +9,6 @@ public import Mathlib.Data.Seq.Basic public import Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized /-! - # Multiseries definitions In this file, we define the multiseries and its main properties: sortedness and approximation. diff --git a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Basic.lean b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Basic.lean index 68198e46873cd0..30623c5de709a9 100644 --- a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Basic.lean +++ b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basis public import Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Predicates /-! - # Computing limits of monomials In this file we define the `Monomial` structure, representing monomials in a basis, i.e. diff --git a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean index 21d3848348f6f1..a36deff28ceace 100644 --- a/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean +++ b/Mathlib/Tactic/ComputeAsymptotics/Multiseries/Monomial/Predicates.lean @@ -8,7 +8,6 @@ module public import Mathlib.Data.Real.Basic /-! - # Predicates on monomials In this file we define `UnitMonomial`: type to represent monomials without coefficient as a list of diff --git a/Mathlib/Tactic/ComputeDegree.lean b/Mathlib/Tactic/ComputeDegree.lean index ef76af3b313f88..6bcf70ea813d23 100644 --- a/Mathlib/Tactic/ComputeDegree.lean +++ b/Mathlib/Tactic/ComputeDegree.lean @@ -8,7 +8,6 @@ module public import Mathlib.Algebra.Polynomial.Degree.Lemmas /-! - # `compute_degree` and `monicity`: tactics for explicit polynomials This file defines two related tactics: `compute_degree` and `monicity`. diff --git a/Mathlib/Tactic/ContinuousFunctionalCalculus.lean b/Mathlib/Tactic/ContinuousFunctionalCalculus.lean index 91e20a30ac9134..8f04da3229b9e3 100644 --- a/Mathlib/Tactic/ContinuousFunctionalCalculus.lean +++ b/Mathlib/Tactic/ContinuousFunctionalCalculus.lean @@ -9,7 +9,6 @@ public import Aesop public import Mathlib.Tactic.FunProp /-! - # Tactics for the continuous functional calculus At the moment, these tactics are just wrappers, but potentially they could be more sophisticated. diff --git a/Mathlib/Tactic/FunProp.lean b/Mathlib/Tactic/FunProp.lean index 5520c3927b2e92..5ebb92a0a3e2a4 100644 --- a/Mathlib/Tactic/FunProp.lean +++ b/Mathlib/Tactic/FunProp.lean @@ -264,7 +264,6 @@ There are four types of theorems that are used a bit differently. arguments, we do not need to add new theorems that addition is also continuous only in the first or only in the second argument. This is automatically inferred using lambda theorems. - - Morphism Theorems: The `fun_prop` tactic can also deal with bundled morphisms. For example, we can state that every continuous linear function is indeed continuous: diff --git a/Mathlib/Tactic/GRewrite.lean b/Mathlib/Tactic/GRewrite.lean index 735224e0c7a539..72a98cc8259a32 100644 --- a/Mathlib/Tactic/GRewrite.lean +++ b/Mathlib/Tactic/GRewrite.lean @@ -8,7 +8,6 @@ module public import Mathlib.Tactic.GRewrite.Elab /-! - # The generalized rewriting tactic The `grw`/`grewrite` tactic is a generalization of the `rewrite` tactic that works with relations diff --git a/Mathlib/Tactic/GRewrite/Elab.lean b/Mathlib/Tactic/GRewrite/Elab.lean index 213dfd28dc98df..28a74aee945dce 100644 --- a/Mathlib/Tactic/GRewrite/Elab.lean +++ b/Mathlib/Tactic/GRewrite/Elab.lean @@ -10,7 +10,6 @@ public meta import Lean.Elab.ConfigEval public import Mathlib.Tactic.GRewrite.Core /-! - # The generalized rewriting tactic This file defines the tactics that use the backend defined in `Mathlib.Tactic.GRewrite.Core`: diff --git a/Mathlib/Tactic/GuardGoalNums.lean b/Mathlib/Tactic/GuardGoalNums.lean index f74462e5f33356..5437e49255b6b2 100644 --- a/Mathlib/Tactic/GuardGoalNums.lean +++ b/Mathlib/Tactic/GuardGoalNums.lean @@ -9,7 +9,6 @@ public import Mathlib.Init public meta import Lean.Elab.Tactic.Basic /-! - A tactic stub file for the `guard_goal_nums` tactic. -/ diff --git a/Mathlib/Tactic/ITauto.lean b/Mathlib/Tactic/ITauto.lean index aea71f5ee3d8d5..df35daaccd6660 100644 --- a/Mathlib/Tactic/ITauto.lean +++ b/Mathlib/Tactic/ITauto.lean @@ -13,7 +13,6 @@ public import Batteries.Tactic.Init public import Mathlib.Util.AtomM /-! - # Intuitionistic tautology (`itauto`) decision procedure The `itauto` tactic will prove any intuitionistic tautology. It implements the well-known diff --git a/Mathlib/Tactic/Linter/TextBased/UnicodeLinter.lean b/Mathlib/Tactic/Linter/TextBased/UnicodeLinter.lean index 8effafd77fbbbf..460eb4d74821a7 100644 --- a/Mathlib/Tactic/Linter/TextBased/UnicodeLinter.lean +++ b/Mathlib/Tactic/Linter/TextBased/UnicodeLinter.lean @@ -9,7 +9,6 @@ import Mathlib.Init /-! - # Tools for the unicode Linter The actual linter is defined in `TextBased.lean`. diff --git a/Mathlib/Tactic/MoveAdd.lean b/Mathlib/Tactic/MoveAdd.lean index 345843b422d91b..854f3f35946f61 100644 --- a/Mathlib/Tactic/MoveAdd.lean +++ b/Mathlib/Tactic/MoveAdd.lean @@ -12,7 +12,6 @@ public meta import Mathlib.Tactic.ToAdditive public meta import Mathlib.Tactic.ToDual /-! - # `move_add` a tactic for moving summands in expressions The tactic `move_add` rearranges summands in expressions. diff --git a/Mathlib/Tactic/NormNum/Prime.lean b/Mathlib/Tactic/NormNum/Prime.lean index adf338a09b55aa..31add80a32f412 100644 --- a/Mathlib/Tactic/NormNum/Prime.lean +++ b/Mathlib/Tactic/NormNum/Prime.lean @@ -16,7 +16,6 @@ public meta import Mathlib.Tactic.NormNum.Result This file provides a `norm_num` extension to prove that natural numbers are prime and compute its minimal factor. Todo: compute the list of all factors. - ## Implementation Notes For numbers larger than 25 bits, the primality proof produced by `norm_num` is an expression diff --git a/Mathlib/Tactic/Polyrith.lean b/Mathlib/Tactic/Polyrith.lean index 267384428fb207..0e4bab31f1020b 100644 --- a/Mathlib/Tactic/Polyrith.lean +++ b/Mathlib/Tactic/Polyrith.lean @@ -9,7 +9,6 @@ public meta import Lean.Elab.Tactic.Basic public import Mathlib.Init /-! - # polyrith Tactic The `polyrith` tactic relied on an external Sage server which has been shut down. diff --git a/Mathlib/Tactic/TFAE.lean b/Mathlib/Tactic/TFAE.lean index 9617a67044616c..603ff81a209f7f 100644 --- a/Mathlib/Tactic/TFAE.lean +++ b/Mathlib/Tactic/TFAE.lean @@ -339,7 +339,6 @@ elab_rules : tactic end Mathlib.Tactic.TFAE /-! - ### Deprecated "Goal-style" `tfae_have` This syntax and its implementation, which behaves like "Mathlib `have`" is deprecated; we preserve diff --git a/Mathlib/Tactic/WLOG.lean b/Mathlib/Tactic/WLOG.lean index f689340b80ff2b..952a2b1f5a2e91 100644 --- a/Mathlib/Tactic/WLOG.lean +++ b/Mathlib/Tactic/WLOG.lean @@ -11,7 +11,6 @@ public import Mathlib.Tactic.Core public import Mathlib.Tactic.Push /-! - # Without loss of generality tactic The tactic `wlog h : P` will add an assumption `h : P` to the main goal, diff --git a/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean b/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean index 3f5841d4654d60..ceade4fa806606 100644 --- a/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean +++ b/Mathlib/Topology/Algebra/Category/ProfiniteGrp/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Algebra.ContinuousMonoidHom public import Mathlib.Topology.Category.Profinite.Basic public import Mathlib.Topology.Separation.Connected /-! - # Category of Profinite Groups We say `G` is a profinite group if it is a topological group which is compact and totally diff --git a/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean b/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean index d123cdc9bed642..bf86697a66619b 100644 --- a/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean +++ b/Mathlib/Topology/Algebra/ContinuousMonoidHom.lean @@ -10,7 +10,6 @@ public import Mathlib.Algebra.Group.Prod public import Mathlib.Topology.Algebra.Group.Defs /-! - # Continuous Monoid Homs This file defines the space of continuous homomorphisms between two topological groups. @@ -290,7 +289,6 @@ end section /-! - ### Continuous MulEquiv This section defines the space of continuous isomorphisms between two topological groups. diff --git a/Mathlib/Topology/Algebra/IsOpenUnits.lean b/Mathlib/Topology/Algebra/IsOpenUnits.lean index 9c84f5cb5ec4cc..18b4efc997b88c 100644 --- a/Mathlib/Topology/Algebra/IsOpenUnits.lean +++ b/Mathlib/Topology/Algebra/IsOpenUnits.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Algebra.GroupWithZero public import Mathlib.Topology.Algebra.Nonarchimedean.AdicTopology /-! - # Topological monoids with open units We say that a topological monoid `M` has open units (`IsOpenUnits`) if `Mˣ` is open in `M` and diff --git a/Mathlib/Topology/Algebra/Localization.lean b/Mathlib/Topology/Algebra/Localization.lean index db7d4bf6e432f6..65d987f9b444c7 100644 --- a/Mathlib/Topology/Algebra/Localization.lean +++ b/Mathlib/Topology/Algebra/Localization.lean @@ -10,7 +10,6 @@ public import Mathlib.RingTheory.OreLocalization.Ring public import Mathlib.Topology.Algebra.Ring.Basic /-! - # Localization of topological rings The topological localization of a topological commutative ring `R` at a submonoid `M` is the ring diff --git a/Mathlib/Topology/Algebra/Module/Compact.lean b/Mathlib/Topology/Algebra/Module/Compact.lean index 4064c08bb9839b..05e93a95fa714d 100644 --- a/Mathlib/Topology/Algebra/Module/Compact.lean +++ b/Mathlib/Topology/Algebra/Module/Compact.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Algebra.Ring.Basic public import Mathlib.RingTheory.Noetherian.Defs /-! - # Compact submodules -/ diff --git a/Mathlib/Topology/Algebra/Module/ModuleTopology.lean b/Mathlib/Topology/Algebra/Module/ModuleTopology.lean index a38676b41761ee..b81ca7af7bc3fb 100644 --- a/Mathlib/Topology/Algebra/Module/ModuleTopology.lean +++ b/Mathlib/Topology/Algebra/Module/ModuleTopology.lean @@ -20,7 +20,6 @@ product topology). Note that `- : A → A` is also automatically continuous as i This topology was suggested by Will Sawin [here](https://mathoverflow.net/a/477763/1384). - ## Mathematical details I (buzzard) don't know of any reference for this other than Sawin's mathoverflow answer, diff --git a/Mathlib/Topology/Algebra/Module/Spaces/CharacterSpace.lean b/Mathlib/Topology/Algebra/Module/Spaces/CharacterSpace.lean index 1b748f50d456cc..66a44d885044a1 100644 --- a/Mathlib/Topology/Algebra/Module/Spaces/CharacterSpace.lean +++ b/Mathlib/Topology/Algebra/Module/Spaces/CharacterSpace.lean @@ -19,7 +19,6 @@ isomorphism between a commutative C⋆-algebra and continuous functions on the c of the algebra. This, in turn, is used to construct the continuous functional calculus on C⋆-algebras. - ## Implementation notes We define `WeakDual.characterSpace 𝕜 A` as a subset of the weak dual, which automatically puts the diff --git a/Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean b/Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean index f2f0b97c865a8a..ba9d18d8e6a30c 100644 --- a/Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean +++ b/Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean @@ -28,7 +28,6 @@ change the associated topology. Finally, it defines `WithIdeal`, a class registering an ideal in a ring and providing the corresponding adic topology to the type class inference system. - ## Main definitions and results * `Ideal.adic_basis`: the basis of submodules given by powers of an ideal. diff --git a/Mathlib/Topology/Algebra/Ring/Basic.lean b/Mathlib/Topology/Algebra/Ring/Basic.lean index 83f775a32f24ea..75a90ff89ae917 100644 --- a/Mathlib/Topology/Algebra/Ring/Basic.lean +++ b/Mathlib/Topology/Algebra/Ring/Basic.lean @@ -13,7 +13,6 @@ public import Mathlib.Topology.Algebra.ContinuousMonoidHom public import Mathlib.Topology.Algebra.Group.GroupTopology /-! - # Topological (semi)rings A topological (semi)ring is a (semi)ring equipped with a topology such that all operations are diff --git a/Mathlib/Topology/Algebra/Ring/Compact.lean b/Mathlib/Topology/Algebra/Ring/Compact.lean index 29ef1a13c75373..64aec7665c0bf7 100644 --- a/Mathlib/Topology/Algebra/Ring/Compact.lean +++ b/Mathlib/Topology/Algebra/Ring/Compact.lean @@ -18,7 +18,6 @@ public import Mathlib.Topology.Algebra.OpenSubgroup public import Mathlib.Topology.Algebra.Ring.Ideal /-! - # Compact Hausdorff Rings ## Main results diff --git a/Mathlib/Topology/Algebra/Valued/ValuativeRel.lean b/Mathlib/Topology/Algebra/Valued/ValuativeRel.lean index 60fce74c07b24e..5f363ce9adaf44 100644 --- a/Mathlib/Topology/Algebra/Valued/ValuativeRel.lean +++ b/Mathlib/Topology/Algebra/Valued/ValuativeRel.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology /-! - # Valuative Relations as Valued In this temporary file, we provide a helper instance diff --git a/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean b/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean index 8247d529eb4b4b..4908607098b494 100644 --- a/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.CompHaus.Limits public import Mathlib.Topology.Category.CompHausLike.EffectiveEpi /-! - # Effective epimorphisms in `CompHaus` This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `CompHaus`. diff --git a/Mathlib/Topology/Category/CompHaus/Limits.lean b/Mathlib/Topology/Category/CompHaus/Limits.lean index 908ec21388ee0a..7d6b38feeec872 100644 --- a/Mathlib/Topology/Category/CompHaus/Limits.lean +++ b/Mathlib/Topology/Category/CompHaus/Limits.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.CompHaus.Basic public import Mathlib.Topology.Category.CompHausLike.Limits /-! - # Explicit limits and colimits This file applies the general API for explicit limits and colimits in `CompHausLike P` (see diff --git a/Mathlib/Topology/Category/CompHausLike/Basic.lean b/Mathlib/Topology/Category/CompHausLike/Basic.lean index 080d42bdc488e6..db9d3ac8d1cac7 100644 --- a/Mathlib/Topology/Category/CompHausLike/Basic.lean +++ b/Mathlib/Topology/Category/CompHausLike/Basic.lean @@ -9,7 +9,6 @@ public import Mathlib.Topology.Category.TopCat.Basic public import Mathlib.CategoryTheory.Functor.EpiMono public import Mathlib.CategoryTheory.Functor.ReflectsIso.Basic /-! - # Categories of Compact Hausdorff Spaces We construct the category of compact Hausdorff spaces satisfying an additional property `P`. diff --git a/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean b/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean index 187b498a02dfc0..2a5a8670de42f7 100644 --- a/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/CompHausLike/EffectiveEpi.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Sites.Coherent.Comparison public import Mathlib.Topology.Category.CompHausLike.Limits /-! - # Effective epimorphisms in `CompHausLike` In any category of compact Hausdorff spaces, continuous surjections are effective epimorphisms. diff --git a/Mathlib/Topology/Category/CompHausLike/Limits.lean b/Mathlib/Topology/Category/CompHausLike/Limits.lean index eedc1febd64392..f07f246d7f2bb8 100644 --- a/Mathlib/Topology/Category/CompHausLike/Limits.lean +++ b/Mathlib/Topology/Category/CompHausLike/Limits.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Extensive public import Mathlib.CategoryTheory.Limits.Preserves.Finite public import Mathlib.Topology.Category.CompHausLike.Basic /-! - # Explicit limits and colimits This file collects some constructions of explicit limits and colimits in `CompHausLike P`, diff --git a/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean b/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean index 7402db74de8d50..47eeabbccb1ce8 100644 --- a/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean +++ b/Mathlib/Topology/Category/CompHausLike/SigmaComparison.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Limits.Shapes.Opposites.Products public import Mathlib.Topology.Category.CompHausLike.Limits /-! - # The sigma-comparison map This file defines the map `CompHausLike.sigmaComparison` associated to a presheaf `X` on diff --git a/Mathlib/Topology/Category/CompactlyGenerated.lean b/Mathlib/Topology/Category/CompactlyGenerated.lean index 1509d32e544470..b3009a41f3204b 100644 --- a/Mathlib/Topology/Category/CompactlyGenerated.lean +++ b/Mathlib/Topology/Category/CompactlyGenerated.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Compactness.CompactlyGeneratedSpace public import Mathlib.CategoryTheory.Elementwise /-! - # Compactly generated topological spaces This file defines the category of compactly generated topological spaces. These are spaces `X` such diff --git a/Mathlib/Topology/Category/Compactum.lean b/Mathlib/Topology/Category/Compactum.lean index 9eacb2c74d35a2..28466d0d02c460 100644 --- a/Mathlib/Topology/Category/Compactum.lean +++ b/Mathlib/Topology/Category/Compactum.lean @@ -13,7 +13,6 @@ public import Mathlib.Topology.Category.Profinite.Basic public import Mathlib.Data.Set.Constructions /-! - # Compacta and Compact Hausdorff Spaces Recall that, given a monad `M` on `Type*`, an *algebra* for `M` consists of the following data: diff --git a/Mathlib/Topology/Category/LightProfinite/Basic.lean b/Mathlib/Topology/Category/LightProfinite/Basic.lean index 51a5db192502cd..4279b32b55f98e 100644 --- a/Mathlib/Topology/Category/LightProfinite/Basic.lean +++ b/Mathlib/Topology/Category/LightProfinite/Basic.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Category.Profinite.AsLimit public import Mathlib.Topology.Category.Profinite.CofilteredLimit public import Mathlib.Topology.ClopenBox /-! - # Light profinite spaces We construct the category `LightProfinite` of light profinite topological spaces. These are diff --git a/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean b/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean index 49fcbbed8cbf51..996ac938819875 100644 --- a/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/LightProfinite/EffectiveEpi.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.CompHausLike.EffectiveEpi public import Mathlib.Topology.Category.LightProfinite.Limits /-! - # Effective epimorphisms in `LightProfinite` This file proves that `EffectiveEpi` and `Surjective` are equivalent in `LightProfinite`. diff --git a/Mathlib/Topology/Category/LightProfinite/Extend.lean b/Mathlib/Topology/Category/LightProfinite/Extend.lean index e9a8de0b273b8f..ef76d5c72793a8 100644 --- a/Mathlib/Topology/Category/LightProfinite/Extend.lean +++ b/Mathlib/Topology/Category/LightProfinite/Extend.lean @@ -9,7 +9,6 @@ public import Mathlib.Topology.Category.LightProfinite.AsLimit public import Mathlib.Topology.Category.Profinite.Extend /-! - # Extending cones in `LightProfinite` Let `(Sₙ)_{n : ℕᵒᵖ}` be a sequential inverse system of finite sets and let `S` be diff --git a/Mathlib/Topology/Category/LightProfinite/Injective.lean b/Mathlib/Topology/Category/LightProfinite/Injective.lean index 061b1b2f7108a1..cc130560b4372c 100644 --- a/Mathlib/Topology/Category/LightProfinite/Injective.lean +++ b/Mathlib/Topology/Category/LightProfinite/Injective.lean @@ -13,7 +13,6 @@ public import Mathlib.CategoryTheory.EpiMono public import Mathlib.Order.RelClasses /-! - # Injectivity of light profinite spaces This file establishes that non-empty light profinite sets are injective in the diff --git a/Mathlib/Topology/Category/LightProfinite/Limits.lean b/Mathlib/Topology/Category/LightProfinite/Limits.lean index 2fc6bc3ce6ee00..0ac53af18d8459 100644 --- a/Mathlib/Topology/Category/LightProfinite/Limits.lean +++ b/Mathlib/Topology/Category/LightProfinite/Limits.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.CompHausLike.Limits public import Mathlib.Topology.Category.LightProfinite.Basic /-! - # Explicit limits and colimits This file applies the general API for explicit limits and colimits in `CompHausLike P` (see diff --git a/Mathlib/Topology/Category/LightProfinite/Sequence.lean b/Mathlib/Topology/Category/LightProfinite/Sequence.lean index a829e49ff1533e..7c283fe1dc9a84 100644 --- a/Mathlib/Topology/Category/LightProfinite/Sequence.lean +++ b/Mathlib/Topology/Category/LightProfinite/Sequence.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Compactification.OnePoint.Basic public import Mathlib.Topology.Category.LightProfinite.Basic /-! - # The light profinite set classifying convergent sequences This file defines the light profinite set `ℕ∪{∞}`, defined as the one point compactification of diff --git a/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean b/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean index 19936289878c59..05d46815198cf4 100644 --- a/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/Profinite/EffectiveEpi.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Category.CompHaus.EffectiveEpi public import Mathlib.Topology.Category.Profinite.Limits public import Mathlib.Topology.Category.Stonean.Basic /-! - # Effective epimorphisms in `Profinite` This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `Profinite`. diff --git a/Mathlib/Topology/Category/Profinite/Extend.lean b/Mathlib/Topology/Category/Profinite/Extend.lean index 335172a81fb214..7e48d542c67196 100644 --- a/Mathlib/Topology/Category/Profinite/Extend.lean +++ b/Mathlib/Topology/Category/Profinite/Extend.lean @@ -9,7 +9,6 @@ public import Mathlib.Topology.Category.Profinite.AsLimit public import Mathlib.Topology.Category.Profinite.CofilteredLimit public import Mathlib.CategoryTheory.Filtered.Final /-! - # Extending cones in `Profinite` Let `(Sᵢ)_{i : I}` be a family of finite sets indexed by a cofiltered category `I` and let `S` be diff --git a/Mathlib/Topology/Category/Profinite/Limits.lean b/Mathlib/Topology/Category/Profinite/Limits.lean index 682d0c70721f8c..0c75f759c104dd 100644 --- a/Mathlib/Topology/Category/Profinite/Limits.lean +++ b/Mathlib/Topology/Category/Profinite/Limits.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.Profinite.Basic public import Mathlib.Topology.Category.CompHausLike.Limits /-! - # Explicit limits and colimits This file applies the general API for explicit limits and colimits in `CompHausLike P` (see diff --git a/Mathlib/Topology/Category/Sequential.lean b/Mathlib/Topology/Category/Sequential.lean index 493885ee672b50..f2fe414175ce7b 100644 --- a/Mathlib/Topology/Category/Sequential.lean +++ b/Mathlib/Topology/Category/Sequential.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Sequences public import Mathlib.Topology.Instances.Discrete public import Mathlib.Topology.Category.TopCat.Basic /-! - # The category of sequential topological spaces We define the category `Sequential` of sequential topological spaces. We follow the usual template diff --git a/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean b/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean index a41162e6601227..eff44f63df6af5 100644 --- a/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/Stonean/EffectiveEpi.lean @@ -9,7 +9,6 @@ public import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular public import Mathlib.Topology.Category.CompHaus.EffectiveEpi public import Mathlib.Topology.Category.Stonean.Limits /-! - # Effective epimorphisms in `Stonean` This file proves that `EffectiveEpi`, `Epi` and `Surjective` are all equivalent in `Stonean`. diff --git a/Mathlib/Topology/Category/Stonean/Limits.lean b/Mathlib/Topology/Category/Stonean/Limits.lean index d997c51f196ad6..b2d684e7855c21 100644 --- a/Mathlib/Topology/Category/Stonean/Limits.lean +++ b/Mathlib/Topology/Category/Stonean/Limits.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Category.CompHausLike.Limits public import Mathlib.Topology.Category.Stonean.Basic /-! - # Explicit limits and colimits This file applies the general API for explicit limits and colimits in `CompHausLike P` (see diff --git a/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean b/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean index 93bd6dc6fc6fdd..a0a85b9f30a14c 100644 --- a/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean +++ b/Mathlib/Topology/Category/TopCat/EffectiveEpi.lean @@ -8,7 +8,6 @@ module public import Mathlib.CategoryTheory.Limits.Shapes.RegularMono public import Mathlib.Topology.Category.TopCat.Limits.Pullbacks /-! - # Effective epimorphisms in `TopCat` This file proves the result `TopCat.effectiveEpi_iff_isQuotientMap`: diff --git a/Mathlib/Topology/Category/TopCat/GrothendieckTopology.lean b/Mathlib/Topology/Category/TopCat/GrothendieckTopology.lean index cf9d1ffe5faa15..d5711baf33fa14 100644 --- a/Mathlib/Topology/Category/TopCat/GrothendieckTopology.lean +++ b/Mathlib/Topology/Category/TopCat/GrothendieckTopology.lean @@ -13,7 +13,6 @@ public import Mathlib.Topology.Category.TopCat.Limits.Pullbacks public import Mathlib.Topology.Category.TopCat.ULift /-! - # The Grothendieck topology on `TopCat` In this file we define the standard Grothendieck topology on `TopCat`. This is the topology diff --git a/Mathlib/Topology/Category/TopCat/Yoneda.lean b/Mathlib/Topology/Category/TopCat/Yoneda.lean index b3ed32d2721d7b..2b5cc2480b3d3c 100644 --- a/Mathlib/Topology/Category/TopCat/Yoneda.lean +++ b/Mathlib/Topology/Category/TopCat/Yoneda.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Limits.Types.Products public import Mathlib.Topology.Category.TopCat.Limits.Products /-! - # Yoneda presheaves on topologically concrete categories This file develops some API for "topologically concrete" categories, defining universe polymorphic diff --git a/Mathlib/Topology/CompactOpen.lean b/Mathlib/Topology/CompactOpen.lean index f691e208772d12..d6d3eef56ab1a1 100644 --- a/Mathlib/Topology/CompactOpen.lean +++ b/Mathlib/Topology/CompactOpen.lean @@ -28,7 +28,6 @@ topological spaces. * `Homeomorph.curry` combines the currying and uncurrying operations into a homeomorphism `C(X × Y, Z) ≃ₜ C(X, C(Y, Z))`. This homeomorphism exists if `X` and `Y` are locally compact. - ## Tags compact-open, curry, function space diff --git a/Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean b/Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean index 4ace9a549bdba9..3a330b7b9b6e16 100644 --- a/Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean +++ b/Mathlib/Topology/Compactification/OnePoint/ProjectiveLine.lean @@ -20,7 +20,6 @@ We construct a set-theoretic equivalence between TODO: Add the extension of this equivalence to a homeomorphism in the case `K = ℝ`, where `OnePoint ℝ` gets the topology of one-point compactification. - ## Main definitions and results * `OnePoint.equivProjectivization` : the equivalence `OnePoint K ≃ ℙ K (Fin 2 → K)`. diff --git a/Mathlib/Topology/Compactification/OnePoint/Sphere.lean b/Mathlib/Topology/Compactification/OnePoint/Sphere.lean index 2c54099bb3920d..7ec92b2f99633a 100644 --- a/Mathlib/Topology/Compactification/OnePoint/Sphere.lean +++ b/Mathlib/Topology/Compactification/OnePoint/Sphere.lean @@ -9,7 +9,6 @@ public import Mathlib.Topology.Compactification.OnePoint.Basic public import Mathlib.Geometry.Manifold.Instances.Sphere /-! - # One-point compactification of Euclidean space is homeomorphic to the sphere. -/ diff --git a/Mathlib/Topology/Connected/Separation.lean b/Mathlib/Topology/Connected/Separation.lean index 2265a89b0a99a8..2542ede009597c 100644 --- a/Mathlib/Topology/Connected/Separation.lean +++ b/Mathlib/Topology/Connected/Separation.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Separation.Hausdorff public import Mathlib.Topology.Connected.TotallyDisconnected /-! - # Separation and (dis)connectedness properties of topological spaces. This file provides an instance `T2Space X` given `TotallySeparatedSpace X`. diff --git a/Mathlib/Topology/DiscreteQuotient.lean b/Mathlib/Topology/DiscreteQuotient.lean index b5045bc4dc69c2..b4db706448efab 100644 --- a/Mathlib/Topology/DiscreteQuotient.lean +++ b/Mathlib/Topology/DiscreteQuotient.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Separation.Regular public import Mathlib.Topology.Connected.TotallyDisconnected /-! - # Discrete quotients of a topological space. This file defines the type of discrete quotients of a topological space, diff --git a/Mathlib/Topology/EMetricSpace/Lipschitz.lean b/Mathlib/Topology/EMetricSpace/Lipschitz.lean index d0ff94b93fc003..19cf66e3246188 100644 --- a/Mathlib/Topology/EMetricSpace/Lipschitz.lean +++ b/Mathlib/Topology/EMetricSpace/Lipschitz.lean @@ -34,7 +34,6 @@ uniformly continuous, and that locally Lipschitz functions are continuous. * `LocallyLipschitzOn f s`: states that `f` is locally Lipschitz on `s`. * `LocallyLipschitz.continuous`: a locally Lipschitz function is continuous. - ## Implementation notes The parameter `K` has type `ℝ≥0`. This way we avoid conjunction in the definition and have diff --git a/Mathlib/Topology/FiberPartition.lean b/Mathlib/Topology/FiberPartition.lean index 56c7a52991016e..6159f1401c1564 100644 --- a/Mathlib/Topology/FiberPartition.lean +++ b/Mathlib/Topology/FiberPartition.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.LocallyConstant.Basic public import Mathlib.Logic.Function.FiberPartition /-! - This file provides some API surrounding `Function.Fiber` (see `Mathlib/Logic/Function/FiberPartition.lean`) in the presence of a topology on the domain of the function. diff --git a/Mathlib/Topology/Homotopy/Equiv.lean b/Mathlib/Topology/Homotopy/Equiv.lean index e2483ec51a00c1..f7620c60d31ffa 100644 --- a/Mathlib/Topology/Homotopy/Equiv.lean +++ b/Mathlib/Topology/Homotopy/Equiv.lean @@ -8,7 +8,6 @@ module public import Mathlib.Topology.Homotopy.Basic /-! - # Homotopy equivalences between topological spaces In this file, we define homotopy equivalences between topological spaces `X` and `Y` as a pair of diff --git a/Mathlib/Topology/JacobsonSpace.lean b/Mathlib/Topology/JacobsonSpace.lean index 8dc924808ef18b..43dba400ab6e31 100644 --- a/Mathlib/Topology/JacobsonSpace.lean +++ b/Mathlib/Topology/JacobsonSpace.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Separation.Regular public import Mathlib.Tactic.CrossRefAttribute /-! - # Jacobson spaces ## Main results diff --git a/Mathlib/Topology/MetricSpace/MetricSeparated.lean b/Mathlib/Topology/MetricSpace/MetricSeparated.lean index 0270e069e35ce0..7689e4f97d5921 100644 --- a/Mathlib/Topology/MetricSpace/MetricSeparated.lean +++ b/Mathlib/Topology/MetricSpace/MetricSeparated.lean @@ -14,7 +14,6 @@ public import Mathlib.Topology.MetricSpace.Antilipschitz This file defines a few notions of separations of sets in a metric space. - The first notion (`Metric.IsSeparated`) is quantitative and describes a single set: a set `s` is `ε`-separated if the distance between any two distinct elements is strictly greater than `ε` diff --git a/Mathlib/Topology/MetricSpace/Ultra/Pi.lean b/Mathlib/Topology/MetricSpace/Ultra/Pi.lean index 2d5b0efd354156..eb5287175dfbcb 100644 --- a/Mathlib/Topology/MetricSpace/Ultra/Pi.lean +++ b/Mathlib/Topology/MetricSpace/Ultra/Pi.lean @@ -17,7 +17,6 @@ This file contains results on the behavior of ultrametrics in products of ultram * `Pi.instIsUltrametricDist`: a product of ultrametric spaces is ultrametric. - ultrametric, nonarchimedean -/ diff --git a/Mathlib/Topology/OpenPartialHomeomorph/Basic.lean b/Mathlib/Topology/OpenPartialHomeomorph/Basic.lean index bb87598e2874a5..f27834ef279bbb 100644 --- a/Mathlib/Topology/OpenPartialHomeomorph/Basic.lean +++ b/Mathlib/Topology/OpenPartialHomeomorph/Basic.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Sets.Opens /-! # Partial homeomorphisms: basic theory - ## Main definitions * `OpenPartialHomeomorph.refl`: the identity open partial homeomorphism diff --git a/Mathlib/Topology/Semicontinuity/Basic.lean b/Mathlib/Topology/Semicontinuity/Basic.lean index 989b1b661f0a15..f2c6aa88e9490a 100644 --- a/Mathlib/Topology/Semicontinuity/Basic.lean +++ b/Mathlib/Topology/Semicontinuity/Basic.lean @@ -52,7 +52,6 @@ ones for lower semicontinuous functions using `OrderDual`. * * - + lower and upper semicontinuity correspond to `r := (f · > ·)` and `r := (f · < ·)`; + lower and upper hemicontinuity correspond to `r := (fun x s ↦ IsOpen s ∧ ((f x) ∩ s).Nonempty)` and `r := (fun x s ↦ s ∈ 𝓝ˢ (f x))`, respectively. diff --git a/Mathlib/Topology/Sheaves/Alexandrov.lean b/Mathlib/Topology/Sheaves/Alexandrov.lean index 128a90ea4a520a..27270c062f53eb 100644 --- a/Mathlib/Topology/Sheaves/Alexandrov.lean +++ b/Mathlib/Topology/Sheaves/Alexandrov.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Order.UpperLowerSetTopology public import Mathlib.Topology.Sheaves.SheafCondition.OpensLeCover /-! - Let `X` be a preorder `≤`, and consider the associated Alexandrov topology on `X`. Given a functor `F : X ⥤ C` to a complete category, we can extend `F` to a presheaf on (the topological space) `X` by taking the right Kan extension along the canonical diff --git a/Mathlib/Topology/Sheaves/LocallySurjective.lean b/Mathlib/Topology/Sheaves/LocallySurjective.lean index 691f46d117301d..d6738602860842 100644 --- a/Mathlib/Topology/Sheaves/LocallySurjective.lean +++ b/Mathlib/Topology/Sheaves/LocallySurjective.lean @@ -12,7 +12,6 @@ public import Mathlib.CategoryTheory.Sites.LocallySurjective public import Mathlib.CategoryTheory.Sites.EpiMono /-! - # Locally surjective maps of presheaves. Let `X` be a topological space, `ℱ` and `𝒢` presheaves on `X`, `T : ℱ ⟶ 𝒢` a map. diff --git a/Mathlib/Topology/Sheaves/SheafCondition/Sites.lean b/Mathlib/Topology/Sheaves/SheafCondition/Sites.lean index 392e27c8a3faf4..faba213792c97f 100644 --- a/Mathlib/Topology/Sheaves/SheafCondition/Sites.lean +++ b/Mathlib/Topology/Sheaves/SheafCondition/Sites.lean @@ -10,7 +10,6 @@ public import Mathlib.Topology.Sheaves.Sheaf public import Mathlib.CategoryTheory.Sites.DenseSubsite.Basic /-! - # Coverings and sieves; from sheaves on sites and sheaves on spaces In this file, we connect coverings in a topological space to sieves in the associated Grothendieck diff --git a/Mathlib/Topology/Spectral/Prespectral.lean b/Mathlib/Topology/Spectral/Prespectral.lean index c259887f6e8c49..4aca7738942eb9 100644 --- a/Mathlib/Topology/Spectral/Prespectral.lean +++ b/Mathlib/Topology/Spectral/Prespectral.lean @@ -11,7 +11,6 @@ public import Mathlib.Topology.Sets.OpenCover public import Mathlib.Topology.Spectral.Hom /-! - # Prespectral spaces In this file, we define prespectral spaces as spaces whose lattice of compact opens forms a basis.