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 |
|---|---|---|---|---|
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | Category.assoc]
section HEq
| Mathlib/CategoryTheory/EqToHom.lean | 276 | 279 |
/-
Copyright (c) 2021 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.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
/-!
# Convex independence
This file defines convex indepen... | protected theorem ConvexIndependent.mono {s t : Set E} (hc : ConvexIndependent 𝕜 ((↑) : t → E))
(hs : s ⊆ t) : ConvexIndependent 𝕜 ((↑) : s → E) :=
hc.comp_embedding (s.embeddingOfSubset t hs)
/-- The range of an injective indexed family of points is convex independent iff that family is. -/
theorem Function.I... | Mathlib/Analysis/Convex/Independent.lean | 97 | 104 |
/-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
/-!
# Bivariate polynomials
This file introduces the notation `R[X][Y]` for the polynomial ring `R[X][X]` in two variables,
and the notat... |
@[simp]
lemma evalEval_add (x y : R) (p q : R[X][Y]) :
| Mathlib/Algebra/Polynomial/Bivariate.lean | 65 | 67 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... |
end List
| Mathlib/Data/List/Rotate.lean | 621 | 622 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | @[simp]
theorem normalize_eq (x : α) : normalize x = x :=
mul_one x
| Mathlib/Algebra/GCDMonoid/Basic.lean | 856 | 859 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.MvPolynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mat... | (show x ∈ span R ↑t by
rw [ht]
exact hx)⟩
theorem fg_of_noetherian [IsNoetherian R A] (S : Subalgebra R A) : S.FG :=
fg_of_fg_toSubmodule (IsNoetherian.noetherian (Subalgebra.toSubmodule S))
theorem fg_of_submodule_fg (h : (⊤ : Submodule R A).FG) : (⊤ : Subalgebra R A).FG :=
| Mathlib/RingTheory/Adjoin/FG.lean | 108 | 115 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | exact ae_eq_zero_of_forall_setIntegral_eq_of_sigmaFinite hfg_int hfg
theorem AEFinStronglyMeasurable.ae_eq_zero_of_forall_setIntegral_eq_zero {f : α → E}
(hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(hf_zero : ∀ s : Set α, MeasurableSet s → μ s < ∞ → ∫ x in s, f x ∂μ = 0)
(hf : A... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 277 | 301 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,150 | 1,156 | |
/-
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 | 259 | 259 | |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.ElementarySubstructures
/-!
# Skolem Functions and Downward Löwenheim–Skolem
## Main Definitions
- `FirstOrder.Language.skolem₁` is a la... | · exact
⟨⟨Sigma.map Nat.succ fun _ => id,
Nat.succ_injective.sigma_map fun _ => Function.injective_id⟩⟩
· refine _root_.trans BoundedFormula.card_le (lift_le.{max u v}.1 ?_)
simp only [mk_empty, lift_zero, lift_uzero, zero_add]
rfl
/-- The structure assigning each function symbol of `L.skol... | Mathlib/ModelTheory/Skolem.lean | 65 | 73 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... |
end Prod
| Mathlib/Order/Filter/Prod.lean | 426 | 428 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... | Iff.intro
(by
rintro hA a b ha hb hab w ⟨au, ⟨u, hu, rfl⟩, bv, ⟨v, hv, rfl⟩, rfl⟩
exact hA hu hv ha hb hab)
fun h _ hx _ hy _ _ ha hb hab => (h ha hb hab) (Set.add_mem_add ⟨_, hx, rfl⟩ ⟨_, hy, rfl⟩)
alias ⟨Convex.set_combo_subset, _⟩ := convex_iff_pointwise_add_subset
| Mathlib/Analysis/Convex/Basic.lean | 76 | 82 |
/-
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... | · exact one_ne_zero
· exact one_ne_zero
section
variable [IsFractionRing B L]
lemma coeSubmodule_differentIdeal [NoZeroSMulDivisors A B] :
| Mathlib/RingTheory/DedekindDomain/Different.lean | 416 | 422 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... | (e'.isImage_source_target_of_disjoint _ hs.symm ht.symm) :=
copy_eq ..
section Pi
variable {ι : Type*} {αi βi γi : ι → Type*}
/-- The product of a family of partial equivalences, as a partial equivalence on the pi type. -/
| Mathlib/Logic/Equiv/PartialEquiv.lean | 840 | 847 |
/-
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... | Set.range (Finsupp.mapRange (α := α) e he₀) = {g | ∀ i, g i ∈ Set.range e} := by
ext g
simp only [Set.mem_range, Set.mem_setOf]
constructor
· rintro ⟨g, rfl⟩ i
simp
· intro h
classical
| Mathlib/Data/Finsupp/Defs.lean | 328 | 335 |
/-
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... |
@[simp]
theorem toIocDiv_sub' (a b : α) : toIocDiv hp (a - p) b = toIocDiv hp a b + 1 := by
| Mathlib/Algebra/Order/ToIntervalMod.lean | 293 | 295 |
/-
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.Algebra.Module.LinearMap.End
import Mathlib.Algebra.Module.Submodule.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Defs
/-!
# Linear maps in... |
/-- If two submodules `p` and `p'` satisfy `p ⊆ p'`, then `inclusion p p'` is the linear map version
of this inclusion. -/
def inclusion (h : p ≤ p') : p →ₗ[R] p' :=
| Mathlib/Algebra/Module/Submodule/LinearMap.lean | 327 | 330 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,218 | 1,219 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | @[simp]
theorem eta : ∀ o : Part α, (⟨o.Dom, fun h => o.get h⟩ : Part α) = o
| ⟨_, _⟩ => rfl
| Mathlib/Data/Part.lean | 79 | 82 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
/-!
# Filters used in box-based integrals
First ... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 530 | 534 | |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | Tendsto (hf.stieltjesFunction a) atTop (𝓝 1) := by
have h_exists : ∀ x : ℝ, ∃ q : ℚ, x - 1 < q ∧ ↑q < x := fun x ↦ exists_rat_btwn (sub_one_lt x)
let qs : ℝ → ℚ := fun x ↦ (h_exists x).choose
have hqs_tendsto : Tendsto qs atTop atTop := by
rw [tendsto_atTop_atTop]
refine fun q ↦ ⟨q + 1, fun y hy ↦ ?_... | Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 380 | 388 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Order.Filter.ZeroAndBoundedAtFilt... | Tendsto UpperHalfPlane.coe atImInfty (comap Complex.im atTop) := by
simpa only [atImInfty, tendsto_comap_iff, Function.comp_def,
funext UpperHalfPlane.coe_im] using tendsto_comap
| Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean | 77 | 80 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | theorem reflect_neg (f : R[X]) (N : ℕ) : reflect N (-f) = -reflect N f := by
rw [neg_eq_neg_one_mul, ← C_1, ← C_neg, reflect_C_mul, C_neg, C_1, ← neg_eq_neg_one_mul]
@[simp]
| Mathlib/Algebra/Polynomial/Reverse.lean | 358 | 361 |
/-
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 | 1,541 | 1,570 | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... | · intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rcases List.length_eq_one_iff.mp (hω.trans h) with ⟨i, rfl⟩
exact ⟨i, cs.wordProd_singleton i⟩
· rintro ⟨i, rfl⟩
exact cs.length_simple i
theorem length_mul_simple_ne (w : W) (i : B) : ℓ (w * s i) ≠ ℓ w := by
intro eq
| Mathlib/GroupTheory/Coxeter/Length.lean | 161 | 169 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Small.Module
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.TensorProduc... | simp [← Module.FinitePresentation.fg_ker_iff e.toLinearMap e.surjective, Submodule.fg_bot]
lemma LinearEquiv.finitePresentation_iff (e : M ≃ₗ[R] N) :
Module.FinitePresentation R M ↔ Module.FinitePresentation R N :=
⟨fun _ ↦ .of_equiv e, fun _ ↦ .of_equiv e.symm⟩
namespace Module.FinitePresentation
variable (... | Mathlib/Algebra/Module/FinitePresentation.lean | 275 | 287 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 1,918 | 1,919 | |
/-
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
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Sub.Basic
impo... |
end CanonicallyOrderedAdd
section Sub
section NonUnitalNonAssocSemiring
variable [NonUnitalNonAssocSemiring R] [PartialOrder R] [CanonicallyOrderedAdd R]
| Mathlib/Algebra/Order/Ring/Canonical.lean | 86 | 93 |
/-
Copyright (c) 2021 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.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
/-!
# Convex independence
This file defines convex indepen... | index type with the original family. -/
theorem ConvexIndependent.comp_embedding {ι' : Type*} (f : ι' ↪ ι) {p : ι → E}
(hc : ConvexIndependent 𝕜 p) : ConvexIndependent 𝕜 (p ∘ f) := by
intro s x hx
rw [← f.injective.mem_set_image]
| Mathlib/Analysis/Convex/Independent.lean | 73 | 77 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 953 | 968 | |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | · simpa [hb, piecewise] using hs _ hb
end Kernel
| Mathlib/Probability/Kernel/Basic.lean | 409 | 411 |
/-
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, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,965 | 1,967 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 525 | 526 | |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | exact (eq_left_or_mem_Ioo_of_mem_Ico (equivIco p a x).2).resolve_left
(hx ∘ ((equivIco p a).symm_apply_apply x).symm.trans ∘ congrArg _)
left_inv' :=
fun x hx ↦ congrArg _ ((equivIco p a).apply_symm_apply ⟨x, Ioo_subset_Ico_self hx⟩)
| Mathlib/Topology/Instances/AddCircle.lean | 267 | 270 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | refine σ.swap_induction_on (by simp) fun π i j hne h_eq ↦ ?_
rw [Equiv.Perm.signAux_mul, Equiv.Perm.sign_mul, h_eq, Equiv.Perm.sign_swap hne,
Equiv.Perm.signAux_swap hne]
| Mathlib/GroupTheory/Perm/Fin.lean | 293 | 295 |
/-
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.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | Multiset.toFinset_map, Finset.card_image_of_injective _ (RingHom.injective _), natSepDegree]
| Mathlib/FieldTheory/SeparableDegree.lean | 371 | 372 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.LeftHomology
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Right Homology of short complexes
In this file, we define the ... | LeftHomologyData.op_p, ← op_comp, LeftHomologyData.f'_i, op_g]
@[reassoc (attr := simp)]
lemma fromOpcycles_op_cyclesOpIso_inv [S.HasRightHomology]:
S.fromOpcycles.op ≫ S.cyclesOpIso.inv = S.op.toCycles := by
dsimp [cyclesOpIso, fromOpcycles]
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 976 | 981 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.PFun
/-!
# `Tendsto` for relations and partial functions
This file generalizes `Filter` definitions from funct... | rw [PFun.preimage_eq]
exact inter_mem (h' s sl₂) h
end Filter
| Mathlib/Order/Filter/Partial.lean | 248 | 251 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 943 | 944 | |
/-
Copyright (c) 2022 Alex Kontorovich and Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Gr... | `μ_𝓕`, of a right-invariant measure `μ` on `G` to a fundamental domain `𝓕`, a set
in the quotient which has `μ_𝓕`-measure zero, also has measure zero under the
folding of `μ` under the quotient. Note that, if `Γ` is infinite, then the folded map
will take the value `∞` on any open set in the quotient! -/
@[t... | Mathlib/MeasureTheory/Measure/Haar/Quotient.lean | 350 | 375 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,351 | 1,352 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Hom.CompTypeclasses
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Notation.Prod
import Mathlib.Algebra.Ring.Act... |
end DistribMulAction
| Mathlib/GroupTheory/GroupAction/Hom.lean | 822 | 823 |
/-
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 | 3,165 | 3,168 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | theorem circumcenter_eq_centroid (s : Simplex ℝ P 1) :
s.circumcenter = Finset.univ.centroid ℝ s.points := by
have hr :
Set.Pairwise Set.univ fun i j : Fin 2 =>
dist (s.points i) (Finset.univ.centroid ℝ s.points) =
dist (s.points j) (Finset.univ.centroid ℝ s.points) := by
intro i hi j hj hij... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 316 | 326 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
import Mathlib.Logic.Function.ULift
/-!
# The Yoneda emb... | lemma map_yonedaEquiv' {X Y : Cᵒᵖ} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj (unop X) ⟶ F)
(g : X ⟶ Y) : F.map g (yonedaEquiv f) = f.app Y g.unop := by
rw [yonedaEquiv_naturality', yonedaEquiv_comp, yonedaEquiv_yoneda_map]
| Mathlib/CategoryTheory/Yoneda.lean | 509 | 511 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... |
theorem image_inversion_sphere_dist_center (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R '' sphere y (dist y c) = insert c (perpBisector c (inversion c R y) : Set P) := by
rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),
| Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 73 | 76 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | Mathlib/Data/Finset/Prod.lean | 405 | 409 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... |
/-- Given `φ : F ⟶ G`, this is the morphism `K ⟶ mappingCone φ` that is constructed
from a cocycle `α : Cochain K F 1` and a cochain `β : Cochain K G 0`
when a suitable cocycle relation is satisfied. -/
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 470 | 473 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
/-!
# Pi
This file contains lemmas which establish bounds on `Real.pi`.
Notably, t... | apply sin_lt
apply div_pos pi_pos
all_goals apply pow_pos; norm_num
refine lt_of_le_of_lt (le_of_eq ?_) this
rw [pow_succ' _ (n + 1), ← mul_assoc, div_mul_cancel₀, mul_comm]; norm_num
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
π < 2 ^ (n + 1) * √(2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n := by
have... | Mathlib/Data/Real/Pi/Bounds.lean | 28 | 37 |
/-
Copyright (c) 2022 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Category.ModuleCat.Projective
import Mathlib.AlgebraicTopology.ExtraDegeneracy
import Mathlib.CategoryTheory.Abelian.Ext
import Mathlib.G... | instance : EnoughProjectives (Rep k G) :=
Rep.equivalenceModuleMonoidAlgebra.enoughProjectives_iff.2 ModuleCat.enoughProjectives
/-- Given a `k`-linear `G`-representation `V`, `Extⁿ(k, V)` (where `k` is a trivial `k`-linear
`G`-representation) is isomorphic to the `n`th cohomology group of `Hom(P, V)`, where `P` is ... | Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean | 626 | 650 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Reid Barton
-/
import Mathlib.Topology.Separation.Regular
/-!
# The shrinking lemma
In this file we prove a few versions of the shrinking lemma. The lemma says that... | exact iff_of_true this.2 (mem_iUnion₂.2 ⟨_, this.1, this.2⟩)
· push_neg at h
refine iff_of_false (h _ ne.some_mem) ?_
simpa only [chainSupCarrier, mem_iUnion₂, not_exists]
| Mathlib/Topology/ShrinkingLemma.lean | 118 | 121 |
/-
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 | 1,420 | 1,450 | |
/-
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 | 427 | 457 | |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | (Functor.associator _ _ _).inv ≫ Γ₂N₂.natTrans ◫ 𝟙 (@N₂ C _ _) = 𝟙 N₂ := by
ext P : 2
dsimp only [NatTrans.comp_app, NatTrans.hcomp_app, Functor.comp_map, Functor.associator,
NatTrans.id_app, Functor.comp_obj]
rw [Γ₂.map_id, N₂.map_id, comp_id, id_comp, id_comp, identity_N₂_objectwise P]
instance : IsI... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 216 | 226 |
/-
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.Submonoid.BigOperators
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod... | /-- The image of a subsemiring along a ring homomorphism is a subsemiring. -/
@[simps coe toSubmonoid]
def map (f : R →+* S) (s : Subsemiring R) : Subsemiring S :=
{ s.toSubmonoid.map (f : R →* S), s.toAddSubmonoid.map (f : R →+ S) with carrier := f '' s }
| Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 117 | 121 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... |
lemma baseChange_polynomialX : (W'.baseChange B).toAffine.polynomialX =
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 842 | 843 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | theorem sup_apply [∀ i, Max (α' i)] (f g : ∀ i, α' i) (i : ι) : (f ⊔ g) i = f i ⊔ g i :=
rfl
theorem sup_def [∀ i, Max (α' i)] (f g : ∀ i, α' i) : f ⊔ g = fun i => f i ⊔ g i :=
rfl
| Mathlib/Order/Lattice.lean | 799 | 804 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Relations holding pairwise
This file develops pairwise relations ... | Mathlib/Data/Set/Pairwise/Basic.lean | 424 | 434 | |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | EMetric.diam (cthickening ε s) ≤ EMetric.diam s + 2 * ε := by
refine diam_le fun x hx y hy => ENNReal.le_of_forall_pos_le_add fun δ hδ _ => ?_
rw [mem_cthickening_iff, ENNReal.ofReal_coe_nnreal] at hx hy
have hε : (ε : ℝ≥0∞) < ε + δ := ENNReal.coe_lt_coe.2 (lt_add_of_pos_right _ hδ)
| Mathlib/Topology/MetricSpace/Thickening.lean | 339 | 342 |
/-
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... | end List
| Mathlib/Logic/Function/Iterate.lean | 231 | 232 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,400 | 2,402 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,379 | 1,380 | |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes, Kevin Buzzard
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Units.Basic
/-!
# Monoid homomorphisms and units
This file allows to... | alias isUnit_of_map_unit := IsUnit.of_map
| Mathlib/Algebra/Group/Units/Hom.lean | 234 | 235 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.UniqueProds.Basic
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Variations on non-zero divisors in `AddMonoidAlgebra`s
This file studie... | as a product of monomials in the supports of `f` and `g` is a product. -/
theorem mul_apply_mul_eq_mul_of_uniqueMul [Mul A] {f g : MonoidAlgebra R A} {a0 b0 : A}
(h : UniqueMul f.support g.support a0 b0) :
(f * g) (a0 * b0) = f a0 * g b0 := by
classical
simp_rw [mul_apply, sum, ← Finset.sum_product']
refi... | Mathlib/Algebra/MonoidAlgebra/NoZeroDivisors.lean | 68 | 79 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathli... | Mathlib/Algebra/Group/Submonoid/Pointwise.lean | 700 | 704 | |
/-
Copyright (c) 2022 Tian Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tian Chen, Mantas Bakšys
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.D... | decide
· rw [← even_iff_two_dvd, Int.not_even_iff_odd]
apply Odd.pow
simp only [← Int.not_even_iff_odd, even_iff_two_dvd, hx, not_false_iff]
theorem Nat.two_pow_sub_pow {x y : ℕ} (hxy : 2 ∣ x - y) (hx : ¬2 ∣ x) {n : ℕ} (hn : Even n) :
emultiplicity 2 (x ^ n - y ^ n) + 1 =
emultiplicity 2 (x + y... | Mathlib/NumberTheory/Multiplicity.lean | 359 | 373 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | subst ha
rfl
| Mathlib/Data/Option/Basic.lean | 151 | 153 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw
-/
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryBiproduct... | (eqToHom h).hom₁ = eqToHom (by subst h; rfl) := by subst h; rfl
lemma Triangle.eqToHom_hom₂ {A B : Triangle C} (h : A = B) :
(eqToHom h).hom₂ = eqToHom (by subst h; rfl) := by subst h; rfl
| Mathlib/CategoryTheory/Triangulated/Basic.lean | 227 | 229 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | exact div_le_one_of_le₀ (abs_real_inner_le_norm x y) (by positivity)
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 577 | 578 |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | @[to_additive]
theorem IsTopologicalGroup.of_comm_of_nhds_one {G : Type u} [CommGroup G] [TopologicalSpace G]
| Mathlib/Topology/Algebra/Group/Basic.lean | 864 | 865 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Stalks
For a presheaf `F` on a topol... | (W : Opens X) (hxW : x ∈ W) (iWU : W ⟶ U) (iWV : W ⟶ V)
{sU : ToType (F.obj (op U))} {sV : ToType (F.obj (op V))}
(ih : F.map iWU.op sU = F.map iWV.op sV) :
F.germ _ x hxU sU = F.germ _ x hxV sV := by
rw [← F.germ_res iWU x hxW, ← F.germ_res iWV x hxW, ConcreteCategory.comp_apply,
| Mathlib/Topology/Sheaves/Stalks.lean | 393 | 397 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | theorem trans_target : (e.trans e').target = e'.target ∩ e'.symm ⁻¹' e.target :=
rfl
| Mathlib/Topology/PartialHomeomorph.lean | 771 | 773 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 591 | 595 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | In other words, it is a type with the following structures: additive commutative monoid
(`AddCommMonoid`), commutative semigroup (`CommSemigroup`), distributive laws (`Distrib`), and
| Mathlib/Algebra/Ring/Defs.lean | 203 | 204 |
/-
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.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 36 | 36 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | s \ s' =ᵐ[μ] t \ t' :=
h.diff h'
| Mathlib/MeasureTheory/OuterMeasure/AE.lean | 189 | 190 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... |
theorem lt_ord_succ_card (o : Ordinal) : o < (succ o.card).ord :=
| Mathlib/SetTheory/Ordinal/Basic.lean | 1,065 | 1,066 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | induction' n using Nat.strong_induction_on with n IH
have IH' : ∀ x : Fin n, P (f x) := fun n => IH n.1 n.2
| Mathlib/Data/Fintype/Basic.lean | 260 | 261 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | Icc_subset_Icc le_rfl h
| Mathlib/Order/Interval/Finset/Basic.lean | 172 | 173 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | rw [GammaIntegral_add_one h1, mul_div_cancel_left₀]
rintro rfl
rw [zero_re] at h1
exact h1.false
· dsimp only [GammaAux]
have : GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1) := by
have hh1 : -(s + 1).re < n := by
rw [Nat.cast_add, Nat.cast_one] at h1
rw [add_re, one_r... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 273 | 284 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.ElementarySubstructures
/-!
# Skolem Functions and Downward Löwenheim–Skolem
## Main Definitions
- `FirstOrder.Language.skolem₁` is a la... | · exact Classical.epsilon_spec (p := fun a => BoundedFormula.Realize φ default
(Fin.snoc (Subtype.val ∘ x) a)) ⟨a, h⟩
· exact S.fun_mem (LHom.sumInr.onFunction φ) ((↑) ∘ x) (by
exact fun i => (x i).2)
/-- Any `L.sum L.skolem₁`-substructure is an elementary `L`-substructure. -/
noncomputable def ele... | Mathlib/ModelTheory/Skolem.lean | 86 | 95 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
import Mathlib.Topology.FiberBundl... | exact Z.coordChange_comp i j k x hx v
/-- The index set of a vector bundle core, as a convenience function for dot notation -/
@[nolint unusedArguments]
def Index := ι
/-- The base space of a vector bundle core, as a convenience function for dot notation -/
@[nolint unusedArguments, reducible]
def Base := B
| Mathlib/Topology/VectorBundle/Basic.lean | 521 | 530 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.Analysis.Normed.Group.NullSubmodule
/-!
# The Fréche... | local postfix:90 "†" => starRingEnd _
theorem innerSL_norm [Nontrivial E] : ‖(innerSL 𝕜 : E →L⋆[𝕜] E →L[𝕜] 𝕜)‖ = 1 :=
show ‖(toDualMap 𝕜 E).toContinuousLinearMap‖ = 1 from LinearIsometry.norm_toContinuousLinearMap _
variable {E 𝕜}
| Mathlib/Analysis/InnerProductSpace/Dual.lean | 93 | 98 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.TensorProduct.Finite
/-!
# The meta properties of... | exact hg.comp hf
theorem finite_respectsIso : RespectsIso @Finite := by
apply finite_stableUnderComposition.respectsIso
intros
exact Finite.of_surjective _ (RingEquiv.toEquiv _).surjective
lemma finite_containsIdentities : ContainsIdentities @Finite := Finite.id
| Mathlib/RingTheory/RingHom/Finite.lean | 34 | 42 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | Mathlib/Order/Bounds/Basic.lean | 1,643 | 1,652 | |
/-
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... | Mathlib/Topology/Bases.lean | 1,050 | 1,054 | |
/-
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.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Data.Fintype.B... | noncommProd t (comm.mono <| subset_of_le <| t.le_add_left s) := by
rcases s with ⟨⟩
rcases t with ⟨⟩
simp
@[to_additive]
lemma noncommProd_induction (s : Multiset α) (comm)
(p : α → Prop) (hom : ∀ a b, p a → p b → p (a * b)) (unit : p 1) (base : ∀ x ∈ s, p x) :
p (s.noncommProd comm) := by
indu... | Mathlib/Data/Finset/NoncommProd.lean | 150 | 163 |
/-
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... | theorem embDomain_eq_zero {f : α ↪ β} {l : α →₀ M} : embDomain f l = 0 ↔ l = 0 :=
(embDomain_injective f).eq_iff' <| embDomain_zero f
| Mathlib/Data/Finsupp/Defs.lean | 418 | 419 |
/-
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.Order.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... | · rw [le_sup_iff, or_iff_right h],
top_sdiff := fun _ => if_congr top_le_iff rfl rfl }
| Mathlib/Order/Heyting/Basic.lean | 966 | 967 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... | fun ⟨_, _, qa, qb⟩ => lt_trans qa qb⟩
| Mathlib/Data/ENNReal/Basic.lean | 582 | 583 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | ∀ I ≠ (⊥ : FractionalIdeal A⁰ (FractionRing A)), I * I⁻¹ = 1
open FractionalIdeal
variable {R A K}
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 242 | 247 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.Grp.EquivalenceGroupAddGroup
import Mathlib.CategoryTheory.ConcreteCategory.EpiMono
import Mathlib.CategoryTheory.Limits.Constructions.Epi... |
variable {A B : Grp.{u}} (f : A ⟶ B)
@[to_additive AddGrp.forget_grp_preserves_mono]
instance forget_grp_preserves_mono : (forget Grp).PreservesMonomorphisms where
preserves f e := by rwa [mono_iff_injective, ← CategoryTheory.mono_iff_injective] at e
@[to_additive AddGrp.forget_grp_preserves_epi]
instance forget_... | Mathlib/Algebra/Category/Grp/EpiMono.lean | 322 | 331 |
/-
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.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
/-!
# Local homeomorphisms
This file defines local homeomorphisms.
## Main definit... | obtain ⟨gf, hgf, he⟩ := hgf x hx
refine ⟨f.symm.trans gf, ⟨f.map_source hxf, ?_⟩, fun y hy ↦ ?_⟩
· apply (f.left_inv hxf).symm ▸ hgf
· change g y = gf (f.symm y)
rw [← he, Function.comp_apply, f.right_inv hy.1]
theorem map_nhds_eq (hf : IsLocalHomeomorphOn f s) {x : X} (hx : x ∈ s) : (𝓝 x).map f = 𝓝 (f x... | Mathlib/Topology/IsLocalHomeomorph.lean | 101 | 109 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | `b` of `a.length`.
An example might be enlightening. Suppose `a = [2, 2, 3, 4, 2]`. It is a composition of
length 5 of 13. The content of the blocks may be represented as `0011222333344`.
Now take `b = [2, 3]` as a composition of `a.length = 5`. It says that the first 2 blocks of `a`
should be merged, and the last 3 b... | Mathlib/Analysis/Analytic/Composition.lean | 937 | 944 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | WeaklyRegular.innerRegular.measure_eq_iSup hU
theorem innerRegular_measurable [WeaklyRegular μ] :
InnerRegularWRT μ IsClosed fun s => MeasurableSet s ∧ μ s ≠ ∞ :=
WeaklyRegular.innerRegular.measurableSet_of_isOpen (fun _ _ h₁ h₂ ↦ h₁.inter h₂.isClosed_compl)
/-- If `s` is a measurable set, a weakly regular me... | Mathlib/MeasureTheory/Measure/Regular.lean | 902 | 908 |
/-
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 | 1,774 | 1,775 | |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... | Mathlib/Order/SupClosed.lean | 110 | 110 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Jakob von Raumer
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Module.End
import Mathlib.CategoryTheory.Endomo... | Mathlib/CategoryTheory/Preadditive/Basic.lean | 161 | 161 | |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.ModelTheory.FinitelyGenerated
import Mathlib.ModelTheory.PartialEquiv
import Mathlib.ModelTheory.Bundled
import Mathlib.Algebra.Order.Arch... |
/-- Any class in the age of a structure has a representative which is a finitely generated
substructure. -/
theorem age.has_representative_as_substructure :
∀ C ∈ Quotient.mk' '' L.age M, ∃ V : {V : L.Substructure M // FG V},
⟦Bundled.mk V⟧ = C := by
rintro _ ⟨N, ⟨N_fg, ⟨N_incl⟩⟩, N_eq⟩
refine N_eq.symm ... | Mathlib/ModelTheory/Fraisse.lean | 170 | 183 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 1,199 | 1,201 |
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