Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
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/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Equiv
/-! # Infinite sums in top... |
protected theorem ContinuousLinearEquiv.map_tsum [T2Space M] [T2Space M₂] {f : ι → M}
(e : M ≃SL[σ] M₂) : e (∑' z, f z) = ∑' z, e (f z) := by
refine symm (e.tsum_eq_iff.mpr ?_)
rw [e.symm_apply_apply _]
end HasSum
section automorphize
| Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 167 | 178 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | RelMap R ![t₁.realize (Sum.elim v xs), t₂.realize (Sum.elim v xs)] := by
rw [Relations.boundedFormula₂, realize_rel, iff_eq_eq]
refine congr rfl (funext (Fin.cases ?_ ?_))
· simp only [Matrix.cons_val_zero]
· simp only [Matrix.cons_val_succ, Matrix.cons_val_fin_one, forall_const]
| Mathlib/ModelTheory/Semantics.lean | 286 | 290 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 702 | 705 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,736 | 2,743 | |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | @[simp] lemma predAbove_last_of_ne_last {i : Fin (n + 2)} (hi : i ≠ last (n + 1)) :
predAbove (last n) i = castPred i hi := by
| Mathlib/Data/Fin/Basic.lean | 1,266 | 1,267 |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.FieldTheory.Separable
/-!
# Separable degree
This file contains basics about the separable degree of a polynom... |
/-- Every irreducible polynomial can be contracted to a separable polynomial. -/
@[stacks 09H0]
theorem _root_.Irreducible.hasSeparableContraction (q : ℕ) [hF : ExpChar F q] {f : F[X]}
| Mathlib/RingTheory/Polynomial/SeparableDegree.lean | 96 | 99 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... | Mathlib/Data/DFinsupp/Order.lean | 326 | 329 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Order.Hom.Lattice
/-!
# Adding complements to a generalized Boolean algebra
This file embeds any generalized Boolean ... | @[simp] lemma lift_le_comp : lift a ≤ comp b ↔ Disjoint a b := ⟨by rintro ⟨_⟩; assumption, LE.sep⟩
| Mathlib/Order/Booleanisation.lean | 138 | 138 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
/-!
# Subsemigroups: membership criteria
In this file we prove various facts about membership in a subsemigroup.
The i... | theorem mul_mem_sup {S T : Subsemigroup M} {x y : M} (hx : x ∈ S) (hy : y ∈ T) : x * y ∈ S ⊔ T :=
mul_mem (mem_sup_left hx) (mem_sup_right hy)
| Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 82 | 84 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,394 | 1,398 | |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... |
@[deprecated (since := "2024-10-26")]
alias embedding_toPullbackDiag := IsEmbedding.toPullbackDiag
lemma Continuous.mapPullback {X₁ X₂ Y₁ Y₂ Z₁ Z₂}
[TopologicalSpace X₁] [TopologicalSpace X₂] [TopologicalSpace Z₁] [TopologicalSpace Z₂]
{f₁ : X₁ → Y₁} {g₁ : Z₁ → Y₁} {f₂ : X₂ → Y₂} {g₂ : Z₂ → Y₂}
{mapX : X₁... | Mathlib/Topology/SeparatedMap.lean | 46 | 54 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... |
variable (R : Type*) [CommSemiring R] {A B C : Type*} [CommRing A] [CommRing B] [CommRing C]
[Algebra A B] [Algebra A C] [Algebra R A] [Algebra R B] [IsScalarTower R A B]
(P : Ideal B) {Q : Ideal C} (p : Ideal A) [Q.LiesOver p] [P.LiesOver p]
(G : Type*) [Group G] [MulSemiringAction G B] [SMulCommClass G A B]
/... | Mathlib/RingTheory/Ideal/Over.lean | 202 | 221 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | exact
s.centerMass_mem_convexHull (fun x hx => hw₀ _ hx) (hw₁.symm ▸ zero_lt_one) fun x hx => hx
| Mathlib/Analysis/Convex/Combination.lean | 374 | 376 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | embDomain f (mapRange g hg p) = mapRange g hg (embDomain f p) := by
ext a
| Mathlib/Data/Finsupp/Defs.lean | 422 | 423 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | have hS0 : ∏ i ∈ s, I i ≠ 0 := Finset.prod_ne_zero_iff.mpr hS'
have hi0 : I i ≠ 0 := hS i (Finset.mem_insert_self i s)
rw [Finset.prod_insert hi, Finset.sum_insert hi, count_mul K v hi0 hS0, hrec hS']
/-- For every `n ∈ ℕ` and every ideal `I`, `val_v(I^n) = n*val_v(I)`. -/
theorem count_pow (n : ℕ) (I : Fr... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 380 | 393 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Module.ULift
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Tactic.Ring
/-!
# The characteristic predicate of tensor product
## Main ... | rw [TensorProduct.lift.tmul]
apply tmul
| add _ _ _ _ =>
rw [map_add]
apply add <;> assumption
lemma IsTensorProduct.of_equiv (e : M₁ ⊗[R] M₂ ≃ₗ[R] M) (he : ∀ x y, e (x ⊗ₜ y) = f x y) :
IsTensorProduct f := by
have : TensorProduct.lift f = e := by
ext x y
simp [he]
simpa [IsTensorProd... | Mathlib/RingTheory/IsTensorProduct.lean | 115 | 127 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,509 | 2,511 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... |
theorem preimage_sigmaMap_sigma {α' : ι' → Type*} (f : ι → ι') (g : ∀ i, α i → α' (f i))
(s : Set ι') (t : ∀ i, Set (α' i)) :
| Mathlib/Data/Set/Sigma.lean | 174 | 176 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
/-!
## Symmetric quivers and arrow reversal
This file co... | /-- The inclusion of a quiver in its symmetrification -/
@[simps]
def of : Prefunctor V (Symmetrify V) where
obj := id
map := Sum.inl
| Mathlib/Combinatorics/Quiver/Symmetric.lean | 150 | 154 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... | exact mul_le_mul_left' hbc' _
protected lemma addLeftMono [AddZeroClass α] [AddLeftMono α]
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 294 | 296 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | exact ⟨i, h2, h1⟩
theorem IsTopologicalBasis.isOpen_iff {s : Set α} {b : Set (Set α)} (hb : IsTopologicalBasis b) :
IsOpen s ↔ ∀ a ∈ s, ∃ t ∈ b, a ∈ t ∧ t ⊆ s := by simp [isOpen_iff_mem_nhds, hb.mem_nhds_iff]
theorem IsTopologicalBasis.of_isOpen_of_subset {s s' : Set (Set α)} (h_open : ∀ u ∈ s', IsOpen u)
| Mathlib/Topology/Bases.lean | 133 | 138 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | Mathlib/Data/List/Zip.lean | 254 | 256 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,284 | 1,297 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | (separableClosure F E).map i = separableClosure F K :=
(map_le_of_algHom i.toAlgHom).antisymm
(fun x h ↦ ⟨_, (map_mem_separableClosure_iff i.symm).2 h, by simp⟩)
| Mathlib/FieldTheory/SeparableClosure.lean | 125 | 128 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | Mathlib/MeasureTheory/Measure/Restrict.lean | 987 | 993 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathli... |
@[simp, reassoc]
theorem braiding_tensor_right (X Y Z : C) :
(β_ X (Y ⊗ Z)).hom =
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by
apply (cancel_epi (α_ X Y Z).hom).1
| Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 93 | 99 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | exact smul_mem _ _ y_mem
theorem map_ne_zero [Nontrivial R] (hI : I ≠ 0) : I.map h ≠ 0 := by
obtain ⟨x, x_ne_zero, hx⟩ := exists_ne_zero_mem_isInteger hI
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 283 | 286 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | Mathlib/Data/Nat/Bitwise.lean | 484 | 485 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs... | [FunLike F G H] [MonoidHomClass F G H]
(f : F) : Function.Injective f ↔ ∀ a, f a = 1 → a = 1 :=
⟨fun h _ => (map_eq_one_iff f h).mp, fun h x y hxy =>
mul_inv_eq_one.1 <| h _ <| by rw [map_mul, hxy, ← map_mul, mul_inv_cancel, map_one]⟩
| Mathlib/Algebra/Group/Hom/Basic.lean | 110 | 113 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Anatole Dedecker, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
... | rw [← hasDerivWithinAt_univ] at *
exact hc.smul hf
nonrec theorem HasStrictDerivAt.smul (hc : HasStrictDerivAt c c' x) (hf : HasStrictDerivAt f f' x) :
| Mathlib/Analysis/Calculus/Deriv/Mul.lean | 92 | 95 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,133 | 1,139 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | funext x y
cases x <;> cases y <;> rfl
| Mathlib/Data/Int/Bitwise.lean | 263 | 265 |
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.RingTheory.DedekindDom... | theorem Int.not_isField : ¬IsField ℤ := fun h =>
Int.not_even_one <|
(h.mul_inv_cancel two_ne_zero).imp fun a => by rw [← two_mul]; exact Eq.symm
| Mathlib/NumberTheory/NumberField/Basic.lean | 47 | 49 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Limits
/-!
# Image-to-kernel comparison maps
Whenever `f : A ⟶ B` and `g : B ⟶ C` satisfy `w : f ≫ g = 0`,
we have `image_le_ker... | -- as it "loses" the information the morphism factors through the image.
theorem factorThruImageSubobject_comp_imageToKernel (w : f ≫ g = 0) :
factorThruImageSubobject f ≫ imageToKernel f g w = factorThruKernelSubobject g f w := by
| Mathlib/Algebra/Homology/ImageToKernel.lean | 68 | 70 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,193 | 1,201 | |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Convexity properties of `rpow`
We prove basic co... | lemma strictConcaveOn_sqrt : StrictConcaveOn ℝ (Set.Ici 0) (√· : ℝ → ℝ) := by
rw [funext Real.sqrt_eq_rpow]
exact strictConcaveOn_rpow (by positivity) (by linarith)
| Mathlib/Analysis/Convex/SpecificFunctions/Pow.lean | 87 | 89 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | rfl
@[simp]
theorem affineSegment_const_vadd_image (x y : P) (v : V) :
(v +ᵥ ·) '' affineSegment R x y = affineSegment R (v +ᵥ x) (v +ᵥ y) :=
affineSegment_image (AffineEquiv.constVAdd R P v : P →ᵃ[R] P) x y
| Mathlib/Analysis/Convex/Between.lean | 49 | 55 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Yongle Hu
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.RingTheory.Ideal.Pointwise
import Mathlib.RingTheory.Ideal.Quot... | Algebra.smul_def _ _
instance Quotient.tower_quotient_map_quotient [Algebra R S] :
IsScalarTower R (R ⧸ p) (S ⧸ map (algebraMap R S) p) :=
IsScalarTower.of_algebraMap_eq fun x => by
rw [Quotient.algebraMap_eq, Quotient.algebraMap_quotient_map_quotient,
Quotient.mk_algebraMap]
end CommRing
section i... | Mathlib/RingTheory/Ideal/Over.lean | 77 | 89 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... | max (blsub.{u, v} a fun a' ha' => f (a' ♯ b) <| nadd_lt_nadd_right ha' b)
(blsub.{u, v} b fun b' hb' => f (a ♯ b') <| nadd_lt_nadd_left hb' a) := by
apply (blsub_le_iff.2 fun i h => _).antisymm (max_le _ _)
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 240 | 242 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | simp only [integral_sin_pow_even, integral_sin_pow_odd] <;>
refine mul_pos (by norm_num [pi_pos]) (prod_pos fun n _ => div_pos ?_ ?_) <;>
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 677 | 678 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,912 | 1,914 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | /-- The bijection between `𝕎 R` and `ℕ → R`, under the assumption that `p` is invertible in `R`.
In `WittVector.ghostEquiv` we upgrade this to an isomorphism of rings. -/
| Mathlib/RingTheory/WittVector/Basic.lean | 204 | 205 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | @[reassoc (attr := simp)]
theorem coprod.map_map {A₁ A₂ A₃ B₁ B₂ B₃ : C} [HasBinaryCoproduct A₁ B₁] [HasBinaryCoproduct A₂ B₂]
[HasBinaryCoproduct A₃ B₃] (f : A₁ ⟶ A₂) (g : B₁ ⟶ B₂) (h : A₂ ⟶ A₃) (k : B₂ ⟶ B₃) :
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 779 | 781 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... | letI := a.totalSpaceTopology
intro z hz
let e : Trivialization F (π F E) :=
a.trivializationOfMemPretrivializationAtlas (a.pretrivialization_mem_atlas z.proj)
refine (e.continuousAt_of_comp_right ?_
((hf z.proj hz).continuousAt (IsOpen.mem_nhds ?_ ?_))).continuousWithinAt
· exact a.mem_base_pretrivial... | Mathlib/Topology/FiberBundle/Basic.lean | 845 | 851 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.Comma.Arrow
/-!
# Commutative squares
This file provide an API for commutative squares in categories.
If `top`, `left`, `right`... | (sq : CommSq f i p g) : HasLift sq ↔ HasLift sq.unop := by
rw [iff, iff]
exact Nonempty.congr (LiftStruct.unopEquiv sq).toFun (LiftStruct.unopEquiv sq).invFun
| Mathlib/CategoryTheory/CommSq.lean | 241 | 243 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₁), l_ih (chain_of_chain_cons hl₁)]
| Mathlib/Data/List/Chain.lean | 86 | 88 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 585 | 589 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... |
theorem Surjective.iterate (Hsurj : Surjective f) (n : ℕ) : Surjective f^[n] :=
Nat.recOn n surjective_id fun _ ihn ↦ ihn.comp Hsurj
| Mathlib/Logic/Function/Iterate.lean | 91 | 93 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... | variable (hST : IsComplement S T) (hHT : IsComplement H T) (hSK : IsComplement S K)
@[simp] theorem equiv_symm_apply (x : S × T) : (hST.equiv.symm x : G) = x.1.1 * x.2.1 := rfl
@[simp]
| Mathlib/GroupTheory/Complement.lean | 447 | 451 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.TensorProduct.Tower
/-!
# Tensor products of Lie modules
Tensor products of Lie modules carry natural Lie ... | { map (f : M →ₗ[R] P) (g : N →ₗ[R] Q) with
map_lie' := fun {x t} => by
simp only [LinearMap.toFun_eq_coe]
| Mathlib/Algebra/Lie/TensorProduct.lean | 125 | 127 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.Order.Conditionall... | simp only [hs.ker, iInter_true, compl_iInter]
exact countable_iUnion fun n ↦ Set.Finite.countable <| h <| hs.mem _
| Mathlib/Order/Filter/Cofinite.lean | 134 | 135 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
/-! # Optional stopping theorem (fair game theorem)
The optional stopping theor... | ENNReal.ofReal (μ[f n]) by
have hadd : ENNReal.ofReal (∫ ω, f n ω ∂μ) =
ENNReal.ofReal
(∫ ω in {ω | ↑ε ≤ (range (n+1)).sup' nonempty_range_succ fun k => f k ω}, f n ω ∂μ) +
ENNReal.ofReal
(∫ ω in {ω | ((range (n+1)).sup' nonempty_range_succ fun k => f k ω) < ↑ε}, f n ω ∂μ) := by
... | Mathlib/Probability/Martingale/OptionalStopping.lean | 141 | 218 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | rw [length_cons]
exact length_pos_of_mem (by assumption)
suffices k + 1 = l.length by simp [this] at hk
rcases l with - | ⟨hd, tl⟩
· simp at hk
· rw [nodup_iff_injective_get] at hl
| Mathlib/Data/List/Cycle.lean | 172 | 177 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | (gc_coinduced_induced f).monotone_l h
@[simp]
theorem induced_top : (⊤ : TopologicalSpace α).induced g = ⊤ :=
(gc_coinduced_induced g).u_top
| Mathlib/Topology/Order.lean | 385 | 389 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Grothendieck
import Mathlib.CategoryTheory.Sites.Pretopology
import Mathlib.CategoryTheory.Limits.Lattice
import Mathlib.Topology.Sets... | end Opens
| Mathlib/CategoryTheory/Sites/Spaces.lean | 92 | 95 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingT... | @[simp]
theorem coeff_zero_eq_zero (hx : IsIntegral A x) : coeff (minpoly A x) 0 = 0 ↔ x = 0 := by
constructor
· intro h
have zero_root := zero_isRoot_of_coeff_zero_eq_zero h
rw [← root hx zero_root]
exact RingHom.map_zero _
· rintro rfl
simp
/-- The minimal polynomial of a nonzero element has no... | Mathlib/FieldTheory/Minpoly/Field.lean | 283 | 298 |
/-
Copyright (c) 2022 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
/-! # Pointwise instances on `AffineSubspace`s
This file provides the additive action `AffineSubspace.po... | (v +ᵥ s).direction = s.direction := by
rw [pointwise_vadd_eq_map, map_direction]
exact Submodule.map_id _
| Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 63 | 66 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 269 | 270 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | /-- In a complete space, if a family of closed sets with diameter tending to `0` is such that each
finite intersection is nonempty, then the total intersection is also nonempty. -/
| Mathlib/Topology/MetricSpace/Bounded.lean | 503 | 504 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 3,265 | 3,267 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | reflTransGen_eq_self reflexive_reflTransGen transitive_reflTransGen
| Mathlib/Logic/Relation.lean | 559 | 560 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Tactic.AdaptationNote
/-!
# Relations
This file defines bundled relations. A relation between `α` and `β` is a f... | r.core_subset
theorem core_inter (s t : Set β) : r.core (s ∩ t) = r.core s ∩ r.core t :=
Set.ext (by simp [mem_core, imp_and, forall_and])
theorem core_union (s t : Set β) : r.core s ∪ r.core t ⊆ r.core (s ∪ t) :=
r.core_mono.le_map_sup s t
@[simp]
theorem core_univ : r.core Set.univ = Set.univ :=
| Mathlib/Data/Rel.lean | 288 | 297 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,405 | 1,411 | |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | simp only [mul_apply, stdBasisMatrix, boole_mul]
by_cases h₁ : i = i'' <;> by_cases h₂ : j = j'' <;> simp [h₁, h₂]
@[simp]
| Mathlib/Data/Matrix/Basis.lean | 241 | 244 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... |
protected alias ⟨_, Nonempty.encard_pos⟩ := encard_pos
@[simp] theorem encard_singleton (e : α) : ({e} : Set α).encard = 1 := by
| Mathlib/Data/Set/Card.lean | 111 | 114 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | section UniqueRepresentative
/-- An auxiliary result en route to `ModularGroup.c_eq_zero`. -/
theorem abs_c_le_one (hz : z ∈ 𝒟ᵒ) (hg : g • z ∈ 𝒟ᵒ) : |g 1 0| ≤ 1 := by
let c' : ℤ := g 1 0
let c := (c' : ℝ)
suffices 3 * c ^ 2 < 4 by
rw [← Int.cast_pow, ← Int.cast_three, ← Int.cast_four, ← Int.cast_mul, Int.c... | Mathlib/NumberTheory/Modular.lean | 443 | 472 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... |
lemma le_traceDual {I J : Submodule B L} :
I ≤ Jᵛ ↔ I * J ≤ 1ᵛ := by
| Mathlib/RingTheory/DedekindDomain/Different.lean | 73 | 75 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
/-!
# Met... |
theorem image_coe_closedBall (z : ℍ) (r : ℝ) :
((↑) : ℍ → ℂ) '' closedBall (α := ℍ) z r = closedBall ↑(z.center r) (z.im * Real.sinh r) := by
ext w; constructor
· rintro ⟨w, hw, rfl⟩
exact dist_le_iff_dist_coe_center_le.1 hw
| Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 275 | 280 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... |
protected theorem fundamentalInterior : IsFundamentalDomain G (fundamentalInterior G s) μ where
nullMeasurableSet := hs.nullMeasurableSet.fundamentalInterior _ _
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 597 | 599 |
/-
Copyright (c) 2024 Yaël Dillies, Kin Yau James Wong. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kin Yau James Wong, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Probability.Kernel.Composition.MeasureCompProd
... | Mathlib/Probability/Kernel/Disintegration/Basic.lean | 399 | 412 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
/-!
... |
See also `Complex.tan_eq_zero_iff` for a version that takes into account junk values of `θ`. -/
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 75 | 76 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.RingTheory.HahnSeries.Multiplication
/-!
# Summable families of Hahn Series
We introduce a notion of formal summability for families of Hahn series, a... | simp [h]
end OfFinsupp
section EmbDomain
variable [PartialOrder Γ] [AddCommMonoid R] {α β : Type*}
open Classical in
/-- A summable family can be reindexed by an embedding without changing its sum. -/
def embDomain (s : SummableFamily Γ R α) (f : α ↪ β) : SummableFamily Γ R β where
toFun b := if h : b ∈ Set.ran... | Mathlib/RingTheory/HahnSeries/Summable.lean | 536 | 553 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.CategoryTheory.ConcreteCategory.Elementwise
import Mathlib.CategoryTheory.Limits.Types.Filter... | change M.mk F _ * M.mk F _ * M.mk F _ = M.mk F _ * M.mk F _
dsimp
rw [colimit_mul_mk_eq F ⟨j₁, x⟩ ⟨j₂, y⟩ (IsFiltered.max j₁ (IsFiltered.max j₂ j₃))
(IsFiltered.leftToMax j₁ (IsFiltered.max j₂ j₃))
(IsFiltered.leftToMax j₂ j₃ ≫ IsFiltered.rightToMax _ _),
colimit_mul_mk_eq ... | Mathlib/Algebra/Category/MonCat/FilteredColimits.lean | 190 | 202 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.GroupRingAction
import Mathlib.Algebra.Ring.Action.Field
import Mathlib.Algebra.Ring.Action.Invariant
import Mathlib.FieldTheory.Finiteness
im... | ← IsInvariantSubring.coe_subtypeHom' G (FixedPoints.subfield G F).toSubring,
← MulSemiringActionHom.coe_polynomial, ← MulSemiringActionHom.map_smul, smul_polynomial,
MulSemiringActionHom.coe_polynomial, IsInvariantSubring.coe_subtypeHom',
Polynomial.eval_map, Subfield.toSubring_subtype_eq_subtype, hf, s... | Mathlib/FieldTheory/Fixed.lean | 202 | 221 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import ... | = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}} := by
ext s
simp only [mem_sym2_iff, Sym2.mem_iff, forall_eq_or_imp, forall_eq, Set.sep_and,
Set.mem_inter_iff, Set.mem_sep_iff, mem_cliqueSet_iff, Set.mem_setOf_eq,
and_and_and_comm (b := _ ∈ _), and_self, is3Clique_iff]
constru... | Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | 90 | 131 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | -- Porting note (https://github.com/leanprover-community/mathlib4/issues/12129): additional beta reduction needed in the middle of the rewrite
rw [orderOf, mul_left_iterate]; beta_reduce; rw [mul_one]
@[to_additive]
| Mathlib/GroupTheory/OrderOfElement.lean | 177 | 180 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | Mathlib/Order/Disjoint.lean | 814 | 815 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | · refine Nat.decreasingInduction' (fun n hn _ ih ↦ ?_) (le_of_not_le h) ?_
· rw [coeff_divByMonic_X_sub_C_rec, ih, eq_comm, Icc_eq_cons_Ioc (Nat.succ_le.mpr hn),
sum_cons, Nat.sub_self, pow_zero, one_mul, mul_sum]
congr 1; refine sum_congr ?_ fun i hi ↦ ?_
· ext; simp [Nat.succ_le]
rw ... | Mathlib/Algebra/Polynomial/Div.lean | 451 | 456 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... |
theorem image_update_Ico_right (f : ∀ i, α i) (i : ι) (b : α i) :
update f i '' Ico (f i) b = Ico f (update f i b) := by simpa using image_update_Ico f i (f i) b
| Mathlib/Order/Interval/Set/Pi.lean | 154 | 157 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... |
lemma hasDerivAt_fourierChar (x : ℝ) : HasDerivAt (𝐞 · : ℝ → ℂ) (2 * π * I * 𝐞 x) x := by
have h1 (y : ℝ) : 𝐞 y = fourier 1 (y : UnitAddCircle) := by
rw [fourierChar_apply, fourier_coe_apply]
push_cast
ring_nf
| Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 93 | 98 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
/-!
# Almost everywhere disjoint sets
We say that sets `s` and `t` are `μ`-a.e. disjoint (see `MeasureTheory.... | (hf : s.PairwiseDisjoint f) : s.Pairwise (AEDisjoint μ on f) :=
hf.mono' fun _i _j h => h.aedisjoint
| Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 66 | 67 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... | variable {R : Type*} [CommRing R] [LinearOrder R] [IsStrictOrderedRing R]
variable {M : Type*} [AddCommGroup M] [Module R M]
variable {ι : Type*}
namespace Orientation
/-- A module `M` over a linearly ordered commutative ring has precisely two "orientations" with
| Mathlib/LinearAlgebra/Orientation.lean | 193 | 199 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | | func f ts ih => simp only [Term.realize, LHom.onTerm, LHom.map_onFunction, ih]
end LHom
@[simp]
| Mathlib/ModelTheory/Semantics.lean | 208 | 212 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 1,033 | 1,035 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 425 | 427 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | /-- See `ENNReal.mul_div_cancel'` for a stronger version. -/
protected lemma mul_div_cancel (ha₀ : a ≠ 0) (ha : a ≠ ∞) : a * (b / a) = b :=
ENNReal.mul_div_cancel' (by simp [ha₀]) (by simp [ha])
| Mathlib/Data/ENNReal/Inv.lean | 169 | 171 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | · exact Or.inr ⟨by rwa [erase_eq_of_not_mem ha], not_mem_erase _ _⟩
@[simp]
theorem memberSubfamily_memberSubfamily : (𝒜.memberSubfamily a).memberSubfamily a = ∅ := by
ext
simp
@[simp]
theorem memberSubfamily_nonMemberSubfamily : (𝒜.nonMemberSubfamily a).memberSubfamily a = ∅ := by
ext
simp
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 99 | 110 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... |
lemma indicator_prod_one {t : Set κ} {j : κ} :
(s ×ˢ t).indicator (1 : ι × κ → M₀) (i, j) = s.indicator 1 i * t.indicator 1 j := by
simp_rw [indicator, mem_prod_eq]
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 62 | 65 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,026 | 1,029 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | @[deprecated (since := "2024-11-19")] alias
isPreirreducible_iff_closed_union_closed := isPreirreducible_iff_isClosed_union_isClosed
/-- A set is irreducible if and only if for every cover by a finite collection of closed sets, it is
contained in one of the members of the collection. -/
theorem isIrreducible_iff_sUnio... | Mathlib/Topology/Irreducible.lean | 251 | 264 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
/-!
# Natural isomorphisms
For the most ... |
theorem naturality_1 (α : F ≅ G) (f : X ⟶ Y) : α.inv.app X ≫ F.map f ≫ α.hom.app Y = G.map f := by
| Mathlib/CategoryTheory/NatIso.lean | 172 | 173 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
@[simp]
theorem IsCycle.zpowersEquivSupport_apply {σ : Perm α} (hσ : IsCycle σ) {n : ℕ} :
hσ.zpowersEquivSupport ⟨σ ^ n, n, rfl⟩ =
⟨(σ ^ n) (Classical.choose hσ),
pow_apply_mem_support.2 (mem_support.2 (Classical.choose_spec hσ).1)⟩ :=
rfl
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 333 | 340 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 318 | 319 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
/-!
# Lemmas about `min` and `max` in an ordered monoid.
-/
op... | theorem max_le_mul_of_one_le [MulLeftMono α] [MulRightMono α] {a b : α} (ha : 1 ≤ a) (hb : 1 ≤ b) :
max a b ≤ a * b :=
max_le_iff.2 ⟨le_mul_of_one_le_right' hb, le_mul_of_one_le_left' ha⟩
end CovariantClassMulLe
| Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 126 | 133 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,792 | 1,801 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
theorem iUnion_Ioc_add_intCast : ⋃ n : ℤ, Ioc (a + n) (a + n + 1) = Set.univ := by
simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using
iUnion_Ioc_add_zsmul zero_lt_one a
| Mathlib/Algebra/Order/ToIntervalMod.lean | 943 | 946 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | · rintro - ⟨U, F, hU, rfl⟩
refine isOpen_biInter_finset fun i hi ↦
(h_basis i).isOpen (t := t i) (hU i hi) |>.mono (iInf_le _ _)
· intro a u ha hu
rcases (nhds_iInf (t := t) (a := a)).symm ▸ hasBasis_iInf'
(fun i ↦ (h_basis i).nhds_hasBasis (t := t i)) |>.mem_iff.1 (hu.mem_nhds ha)
| Mathlib/Topology/Bases.lean | 531 | 536 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,162 | 1,217 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... |
theorem div_eq_one_iff {a b : ℝ≥0∞} (hb₀ : b ≠ 0) (hb₁ : b ≠ ∞) : a / b = 1 ↔ a = b :=
| Mathlib/Data/ENNReal/Inv.lean | 467 | 468 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.SuccPred.Archimedean
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Successor and predecessor limits
We define the pre... | protected theorem IsPredLimit.isMin (h : IsPredLimit (pred a)) : IsMin a :=
h.dual.isMax
theorem not_isPredPrelimit_pred_of_not_isMin (ha : ¬ IsMin a) : ¬ IsPredPrelimit (pred a) :=
mt IsPredPrelimit.isMin ha
theorem not_isPredLimit_pred_of_not_isMin (ha : ¬ IsMin a) : ¬ IsPredLimit (pred a) :=
mt IsPredLimit.i... | Mathlib/Order/SuccPred/Limit.lean | 437 | 445 |
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