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) 2023 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, Ruben Van de Velde
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
impor... | {n : ℕ} {x : 𝕜} {s : Set 𝕜} (hx : x ∈ s) (h : UniqueDiffOn 𝕜 s) {f g : 𝕜 → F}
section
theorem iteratedDerivWithin_congr (hfg : Set.EqOn f g s) :
| Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 24 | 28 |
/-
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... | theorem Ioo_insert_right (h : a < b) : insert b (Ioo a b) = Ioc a b := by
rw [← coe_inj, coe_insert, coe_Ioo, coe_Ioc, Set.insert_eq, Set.union_comm, Set.Ioo_union_right h]
| Mathlib/Order/Interval/Finset/Basic.lean | 591 | 592 |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... | (add : ∀ ⦃f g⦄, ∀ hf : MemLp f p μ, ∀ hg : MemLp g p μ, Disjoint (support f) (support g) →
motive (hf.toLp f) → motive (hg.toLp g) → motive (hf.toLp f + hg.toLp g))
| Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 820 | 821 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | let ⟨γ, pl, ple⟩ := IH dl
⟨β ⊕ γ, p.map (inl ⊗ inr ∘ inl)::pl.map fun q => q.map (inl ⊗ inr ∘ inr),
fun v => by
simp; exact
Iff.trans (and_congr (pe v) (ple v))
⟨fun ⟨⟨m, hm⟩, ⟨n, hn⟩⟩ =>
⟨m ⊗ n, by
rw [show (v ⊗ m ⊗ n) ∘ (inl ⊗ inr ∘ inl) = v ⊗ m from
... | Mathlib/NumberTheory/Dioph.lean | 309 | 317 |
/-
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.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | cast_injective.comp_left.injOn (Set.subset_univ _) ?_
refine (threeAPFree_sphere 0 (√↑k)).mono (Set.image_subset_iff.2 fun x => ?_)
rw [Set.mem_preimage, mem_sphere_zero_iff_norm]
exact norm_of_mem_sphere
theorem threeAPFree_image_sphere :
ThreeAPFree ((sphere n d k).image (map (2 * d - 1)) : Set ℕ) := b... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 176 | 185 |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
/-!
# Infimum separation
This file defines the extended infimum separation of a set. This is approximately dual to the
diame... | ⟨w, hxy, hed⟩
simp_rw [mem_toFinset] at hxy
exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
theorem Finite.infsep_exists_of_nontrivial (hsf : s.Finite) (hs : s.Nontrivial) :
∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y :=
letI := hsf.fintype
hs.infsep_exists_of_finite
end PseudoMetricSpace
... | Mathlib/Topology/MetricSpace/Infsep.lean | 439 | 450 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | hf.desc (CokernelCofork.ofπ (φ.right ≫ cc'.π) (by
erw [← Arrow.w_assoc φ, condition, comp_zero]))
@[reassoc (attr := simp)]
lemma π_mapOfIsColimit {cc : CokernelCofork f} (hf : IsColimit cc) (cc' : CokernelCofork f')
(φ : Arrow.mk f ⟶ Arrow.mk f') :
cc.π ≫ mapOfIsColimit hf cc' φ = φ.right ≫ cc'.π :=
h... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 698 | 707 |
/-
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.Decomposition
import Mathlib.Tactic.FinCases
/-!
# Behaviour of P_infty with respect to degeneracies
For any `X : SimplicialObject C... | induction' q with q hq
· intro i (hi : n + 1 ≤ i)
omega
· intro i (hi : n + 1 ≤ i + q + 1)
by_cases h : n + 1 ≤ (i : ℕ) + q
· rw [P_succ, HomologicalComplex.comp_f, ← assoc, hq i h, zero_comp]
· replace hi : n = i + q := by
obtain ⟨j, hj⟩ := le_iff_exists_add.mp hi
rw [← Nat.lt_suc... | Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean | 57 | 115 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... |
theorem basicOpen_eq_union_of_projection (f : A) :
basicOpen 𝒜 f = ⨆ i : ℕ, basicOpen 𝒜 (GradedAlgebra.proj 𝒜 i f) :=
TopologicalSpace.Opens.ext <|
Set.ext fun z => by
rw [mem_coe_basicOpen, mem_coe, iSup, TopologicalSpace.Opens.mem_sSup]
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 370 | 375 |
/-
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.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· contrapose! h
| Mathlib/Order/SuccPred/Basic.lean | 894 | 895 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Cast.Field
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.CardComm... | exact Nat.card_congr ⟨fun x a => ⟨⟨x.1.1 a, x.1.2 a⟩, x.2 a⟩, fun x => ⟨⟨fun a => (x a).1.1,
fun a => (x a).1.2⟩, fun a => (x a).2⟩, fun x => rfl, fun x => rfl⟩
theorem commProb_function {α β : Type*} [Fintype α] [Mul β] :
commProb (α → β) = (commProb β) ^ Fintype.card α := by
rw [commProb_pi, Finset.prod_... | Mathlib/GroupTheory/CommutingProbability.lean | 54 | 60 |
/-
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.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 959 | 963 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.List.Pairwise
import Mathlib.Logic.Pairwise
import Mathlib.Logic.Relation
/-!
# Pairwise relations on a list
This file provides basic results ... | Mathlib/Data/List/Pairwise.lean | 159 | 167 | |
/-
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.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | variable {X Y Z : LocallyRingedSpace} (f : X ⟶ Z) (g : Y ⟶ Z)
variable [H : LocallyRingedSpace.IsOpenImmersion f]
instance (priority := 100) of_isIso [IsIso g] : LocallyRingedSpace.IsOpenImmersion g :=
@PresheafedSpace.IsOpenImmersion.ofIsIso _ _ _ _ g.1
⟨⟨(inv g).1, by
erw [← LocallyRingedSpace.comp_toS... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 941 | 960 |
/-
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.... | {xs : Fin (m + n) → M} :
(φ.relabel g).Realize v xs ↔
φ.Realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAdd m) := by
apply realize_mapTermRel_add_castLe <;> simp
theorem realize_liftAt {n n' m : ℕ} {φ : L.BoundedFormula α n} {v : α → M} {xs : Fin (n + n') → M}
(hmn : m + n' ≤ n + 1) :
... | Mathlib/ModelTheory/Semantics.lean | 359 | 373 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Jireh Loreaux
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Data.Set... | ⟨fun h =>
Set.ext fun y => ⟨fun ⟨x, hx⟩ => by simp [← hx, h x], fun hy => ⟨0, by simpa using hy.symm⟩⟩,
fun h x => Set.mem_singleton_iff.mp (h ▸ Set.mem_range_self x)⟩
end
| Mathlib/Algebra/Ring/Hom/Basic.lean | 35 | 39 |
/-
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.Order.Disjoint
import Mathlib.Order.RelIso.Basic
import Mathlib.Tactic.Monotonicity.Attr
/-!
# Order homomorphisms
This file defines order homomorphi... | Mathlib/Order/Hom/Basic.lean | 1,278 | 1,281 | |
/-
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.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | @[simp]
theorem some_zero [Zero M] : (0 : Option α →₀ M).some = 0 := by
ext
simp
@[simp]
theorem some_add [AddZeroClass M] (f g : Option α →₀ M) : (f + g).some = f.some + g.some := by
ext
| Mathlib/Data/Finsupp/Basic.lean | 723 | 730 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,084 | 1,085 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | end EdistLeGeometric
section EdistLeGeometricTwo
| Mathlib/Analysis/SpecificLimits/Basic.lean | 431 | 434 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.Field.ZMod
import Mathlib.Data.Nat.Prime.In... | open Polynomial
/-- The cardinality of a field is at most `n` times the cardinality of the image of a degree `n`
polynomial -/
theorem card_image_polynomial_eval [DecidableEq R] [Fintype R] {p : R[X]} (hp : 0 < p.degree) :
Fintype.card R ≤ natDegree p * #(univ.image fun x => eval x p) :=
Finset.card_le_mul_car... | Mathlib/FieldTheory/Finite/Basic.lean | 65 | 72 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.FinRange
import Mathlib.Data.List.Perm.Basic
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Induc... |
@[simp 900]
theorem sublists'_cons (a : α) (l : List α) :
sublists' (a :: l) = sublists' l ++ map (cons a) (sublists' l) := by
simp [sublists'_eq_sublists'Aux, foldr_cons, sublists'Aux_eq_map]
| Mathlib/Data/List/Sublists.lean | 61 | 66 |
/-
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.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Infini... |
theorem summable_div_const_iff (h : a ≠ 0) : (Summable fun i ↦ f i / a) ↔ Summable f := by
| Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 89 | 90 |
/-
Copyright (c) 2023 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.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | lemma rel_symm : Symmetric (Rel t) := fun x y h ↦ by cases h <;> constructor <;> assumption
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 60 | 60 |
/-
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... |
theorem not_isSuccPrelimit_iff' : ¬ IsSuccPrelimit a ↔ a ∈ range (succ : α → α) := by
simp_rw [isSuccPrelimit_iff_succ_ne, not_forall, not_ne_iff, mem_range]
| Mathlib/Order/SuccPred/Limit.lean | 237 | 239 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | @[simp] lemma prod_zero : μ.prod (0 : FiniteMeasure β) = 0 := by
rw [← mass_zero_iff, mass_prod, zero_mass, mul_zero]
| Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 67 | 68 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | Mathlib/RingTheory/OreLocalization/Basic.lean | 735 | 737 | |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | theorem ite_apply (f g : ∀ a, σ a) (a : α) : (ite P f g) a = ite P (f a) (g a) :=
dite_apply P (fun _ ↦ f) (fun _ ↦ g) a
| Mathlib/Logic/Basic.lean | 899 | 900 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... | The subgraph `M` of `G` is a matching if every vertex of `M` is incident to exactly one edge in `M`.
We say that the vertices in `M.support` are *matched* or *saturated*.
-/
def IsMatching (M : Subgraph G) : Prop := ∀ ⦃v⦄, v ∈ M.verts → ∃! w, M.Adj v w
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 63 | 67 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Junyan Xu
-/
import Mathlib.LinearAlgebra.TensorProduct.RightExactness
import Mathlib.LinearAlgebra.TensorProduct.Finiteness
import Mathlib.LinearAlgebra.DirectSum.Finsupp
... | That is, $\sum_i m_i \otimes n_i = 0$. -/
theorem sum_tmul_eq_zero_of_vanishesTrivially (hmn : VanishesTrivially R m n) :
∑ i, m i ⊗ₜ n i = (0 : M ⊗[R] N) := by
obtain ⟨k, a, y, h₁, h₂⟩ := hmn
simp_rw [h₁, tmul_sum, tmul_smul]
rw [Finset.sum_comm]
simp_rw [← tmul_smul, ← smul_tmul, ← sum_tmul, h₂, zero_tmul... | Mathlib/LinearAlgebra/TensorProduct/Vanishing.lean | 102 | 157 |
/-
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 | 1,905 | 1,906 | |
/-
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.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 231 | 248 | |
/-
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.AlgebraicGeometry.ProjectiveSpectrum.Topology
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalizatio... |
/-- `Proj` of a graded ring as a `LocallyRingedSpace` -/
def Proj.toLocallyRingedSpace : LocallyRingedSpace :=
{ Proj.toSheafedSpace 𝒜 with
isLocalRing := fun x =>
@RingEquiv.isLocalRing _ _ _ (show IsLocalRing (at x) from inferInstance) _
(Proj.stalkIso' 𝒜 x).symm }
end
end AlgebraicGeometry
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/StructureSheaf.lean | 352 | 363 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
/-!
# Pullbacks and pushouts in the category of topological spaces
-... | theorem coequalizer_isOpen_iff (U : Set ((coequalizer f g :) : Type u)) :
IsOpen U ↔ IsOpen (coequalizer.π f g ⁻¹' U) :=
isOpen_iff_of_isColimit_cofork _ (coequalizerIsCoequalizer f g) _
end
end TopCat
| Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 423 | 440 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | lemma factorsThru_of_le {X : C} (S T : Presieve X) (h : S ≤ T) :
S.FactorsThru T :=
fun Y g hg => ⟨Y, 𝟙 _, g, h _ hg, by simp⟩
| Mathlib/CategoryTheory/Sites/Coverage.lean | 86 | 88 |
/-
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.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | refine
((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass
fun s _ => ?_
refine ((ordConnected_biInter ?_).inter (ordConnected_biInter ?_)).out <;> intro _ _
| Mathlib/Topology/Order/Basic.lean | 120 | 123 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 561 | 567 | |
/-
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.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | measurableSet_cut (fun _ b => b ∈ s) f fun b => MeasurableSet.const (b ∈ s)
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 161 | 162 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 1,189 | 1,190 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... |
variable {X : Ω → ℝ}
nonrec theorem _root_.Real.hasPDF_iff [SFinite ℙ] :
HasPDF X ℙ ↔ AEMeasurable X ℙ ∧ map X ℙ ≪ volume := by
| Mathlib/Probability/Density.lean | 261 | 265 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | · exact index_pos K₀ hU
@[to_additive]
theorem prehaar_mono {K₀ : PositiveCompacts G} {U : Set G} (hU : (interior U).Nonempty)
{K₁ K₂ : Compacts G} (h : (K₁ : Set G) ⊆ K₂.1) :
prehaar (K₀ : Set G) U K₁ ≤ prehaar (K₀ : Set G) U K₂ := by
simp only [prehaar]; rw [div_le_div_iff_of_pos_right]
· exact mod_cas... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 273 | 283 |
/-
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.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
/-!
# Closure, interior, and frontier of preimages under `re`... |
@[simp]
| Mathlib/Analysis/Complex/ReImTopology.lean | 139 | 140 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... |
theorem le_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (μ : OuterMeasure X) (r : ℝ≥0∞) (h0 : 0 < r)
(hr : ∀ s, diam s ≤ r → μ s ≤ m (diam s)) : μ ≤ mkMetric m :=
le_iSup₂_of_le r h0 <| mkMetric'.le_pre.2 fun _ hs => hr _ hs
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 391 | 394 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | eLpNorm (s.indicator f - t.indicator f) p μ = eLpNorm ((s ∆ t).indicator f) p μ :=
eLpNorm_congr_norm_ae <| ae_of_all _ fun x ↦ by simp [Set.apply_indicator_symmDiff norm_neg]
@[simp]
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 477 | 480 |
/-
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.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | (withDensity_absolutelyContinuous (κ := η) (rnDeriv κ η) a)
lemma singularPart_eq_zero_iff_measure_eq_zero (κ η : Kernel α γ)
[IsFiniteKernel κ] [IsFiniteKernel η] (a : α) :
singularPart κ η a = 0 ↔ κ a (mutuallySingularSetSlice κ η a) = 0 := by
have h_eq_add := rnDeriv_add_singularPart κ η
simp_rw [Ke... | Mathlib/Probability/Kernel/RadonNikodym.lean | 433 | 442 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... |
theorem convexBodySum_isBounded : Bornology.IsBounded (convexBodySum K B) := by
classical
refine Metric.isBounded_iff.mpr ⟨B + B, fun x hx y hy => ?_⟩
refine le_trans (norm_sub_le x y) (add_le_add ?_ ?_)
· exact le_trans (norm_le_convexBodySumFun x) hx
· exact le_trans (norm_le_convexBodySumFun y) hy
theore... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 349 | 360 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... |
/-- `R⸨X⸩` is notation for `LaurentSeries R`. -/
scoped notation:9000 R "⸨X⸩" => LaurentSeries R
| Mathlib/RingTheory/LaurentSeries.lean | 106 | 108 |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 1,160 | 1,170 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | /-- Compose a linear map with a `LinearPMap` -/
def compPMap (g : F →ₗ[R] G) (f : E →ₗ.[R] F) : E →ₗ.[R] G where
domain := f.domain
toFun := g.comp f.toFun
@[simp]
theorem compPMap_apply (g : F →ₗ[R] G) (f : E →ₗ.[R] F) (x) : g.compPMap f x = g (f x) :=
rfl
end LinearMap
namespace LinearPMap
/-- Restrict codo... | Mathlib/LinearAlgebra/LinearPMap.lean | 606 | 639 |
/-
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 | 997 | 1,007 | |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhangir Azerbayev, Adam Topaz, Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Basic
import Mathlib.LinearAlgebra.Alternating.Basic
/-!
# Exterior Algebras
We construct the e... | Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean | 237 | 237 | |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Co... | eq_div_of_mul_eq h2 <| by rwa [mul_comm] at h
theorem add_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
| Mathlib/Tactic/CancelDenoms/Core.lean | 50 | 52 |
/-
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 | 2,107 | 2,109 | |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | @[simp] theorem wordProd_singleton (i : B) : π ([i]) = s i := by simp [wordProd]
theorem wordProd_concat (i : B) (ω : List B) : π (ω.concat i) = π ω * s i := by simp [wordProd]
| Mathlib/GroupTheory/Coxeter/Basic.lean | 366 | 369 |
/-
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.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | theorem natDegree_mul_C_eq_of_mul_eq_one {ai : R} (au : a * ai = 1) :
(p * C a).natDegree = p.natDegree :=
le_antisymm (natDegree_mul_C_le p a)
(calc
p.natDegree = (p * 1).natDegree := by nth_rw 1 [← mul_one p]
_ = (p * C a * C ai).natDegree := by rw [← C_1, ← au, RingHom.map_mul, ← mul_assoc]
... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 111 | 117 |
/-
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.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... |
variable {p q : ℚ≥0}
| Mathlib/Data/NNRat/Defs.lean | 314 | 316 |
/-
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.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Sites.CoverPreserving
/-! Localization
In this file, given a Grothendieck topology `J` on a cat... |
lemma overEquiv_le_overEquiv_iff {X : C} {Y : Over X} (R₁ R₂ : Sieve Y) :
R₁.overEquiv Y ≤ R₂.overEquiv Y ↔ R₁ ≤ R₂ := by
refine ⟨fun h ↦ ?_, fun h ↦ Sieve.functorPushforward_monotone _ _ h⟩
replace h : (overEquiv Y).symm (R₁.overEquiv Y) ≤ (overEquiv Y).symm (R₂.overEquiv Y) :=
Sieve.functorPullback_monot... | Mathlib/CategoryTheory/Sites/Over.lean | 68 | 81 |
/-
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.SetTheory.Cardinal.ENat
/-!
# Projection from cardinal numbers to natural numbers
In this file we define `Cardinal.toNat` to be the natural projectio... |
theorem toNat_mul (x y : Cardinal) : toNat (x * y) = toNat x * toNat y := map_mul toNat x y
| Mathlib/SetTheory/Cardinal/ToNat.lean | 145 | 146 |
/-
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.Algebra.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Denominators of evaluation of polynomials ... | simp only [eval_zero, RingHom.map_zero, mul_zero, Polynomial.map_zero]⟩
theorem denomsClearable_C_mul_X_pow {N : ℕ} (a : R) (bu : bi * i b = 1) {n : ℕ} (r : R)
| Mathlib/Algebra/Polynomial/DenomsClearable.lean | 42 | 44 |
/-
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.Algebra.Ring.Pointwise.Set
import Mathlib.Order.Filter.AtTopBot.CompleteLattice
import Mathlib.Order.Filter.AtTopBot.G... |
@[deprecated (since := "2024-12-22")]
alias nhdsWithin_Ioi_eq_bot_iff := nhdsGT_eq_bot_iff
| Mathlib/Topology/Order/LeftRightNhds.lean | 99 | 102 |
/-
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.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... |
@[simp]
theorem image_openSegment (f : E →ᵃ[𝕜] F) (a b : E) :
f '' openSegment 𝕜 a b = openSegment 𝕜 (f a) (f b) :=
Set.ext fun x => by
| Mathlib/Analysis/Convex/Segment.lean | 214 | 218 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Data.Nat.EvenOddRec
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.LinearCombination
/-!
# Elliptic divisibility sequences
... | preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * (if Even m then b else 1) -
preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * (if Even m then 1 else b) := by
rw [show 2 * (m + 2) + 1 = 2 * m + 5 by rfl, preNormEDS', dif_pos <| even_two_mul _]
| Mathlib/NumberTheory/EllipticDivisibilitySequence.lean | 187 | 189 |
/-
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 | 3,466 | 3,476 | |
/-
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.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 512 | 513 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Submo... |
@[to_additive]
| Mathlib/Algebra/Group/Submonoid/Basic.lean | 325 | 326 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | · dsimp [ZMod, ZMod.cast]
rw [Int.cast_sub, Int.cast_one]
· dsimp [ZMod, ZMod.cast, ZMod.val]
rw [Fin.coe_sub_one, if_neg]
| Mathlib/Data/ZMod/Basic.lean | 541 | 544 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Ahmad Alkhalawi
-/
import Mathlib.Data.Matrix.ConjTranspose
import Mathlib.Tactic.Abel
/-! # Extra lemmas about invertible matrices
A few of the `Invertible` lemmas general... | protected theorem mul_invOf_cancel_left (A : Matrix n n α) (B : Matrix n m α) [Invertible A] :
A * (⅟ A * B) = B := by rw [← Matrix.mul_assoc, mul_invOf_self, Matrix.one_mul]
| Mathlib/Data/Matrix/Invertible.lean | 40 | 41 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | lemma exact_of_g_is_cokernel (hS : IsColimit (CokernelCofork.ofπ S.g S.zero))
[S.HasHomology] : S.Exact := by
rw [exact_iff_mono_fromOpcycles]
have : IsSplitMono S.fromOpcycles :=
⟨⟨{ retraction := hS.desc (CokernelCofork.ofπ S.pOpcycles S.f_pOpcycles)
id := by
rw [← cancel_epi S.pOpcycles... | Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 415 | 423 |
/-
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... | monomial (∑ i ∈ s, f i) a = C a * ∏ i ∈ s, monomial (f i) 1 := by
rw [← monomial_sum_one, C_mul', ← (monomial _).map_smul, smul_eq_mul, mul_one]
| Mathlib/Algebra/MvPolynomial/Basic.lean | 330 | 331 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | natCast_ne_top 1
@[simp]
theorem ofNat_ne_top (x : ℕ) [x.AtLeastTwo] : (ofNat(x) : PartENat) ≠ ⊤ :=
| Mathlib/Data/Nat/PartENat.lean | 339 | 342 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | f (coeff p i * s ^ (p.natDegree - i)) * (f s * a) ^ i := by
simp [eval₂_eq_sum, sum_def]
_ = p.support.sum fun i => f (coeff p i * s ^ (p.natDegree - i)) * (f s * a) ^ i :=
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 124 | 126 |
/-
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.Logic.Denumerable
/-!
# Equivalences involving `List`-like types
This file defines some additional constructive equivalences using `Encodable` and th... | Mathlib/Logic/Equiv/List.lean | 300 | 302 | |
/-
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.Finset.Max
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Multiset.Sort
import Mathlib.Order.RelIso.Set
/-!
# Construct a sorted list f... | rfl
theorem orderEmbOfFin_apply (s : Finset α) {k : ℕ} (h : s.card = k) (i : Fin k) :
s.orderEmbOfFin h i = (s.sort (· ≤ ·))[i]'(by rw [length_sort, h]; exact i.2) :=
| Mathlib/Data/Finset/Sort.lean | 175 | 178 |
/-
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... | haveI := hc.toEncodable
simp only [biUnion_eq_iUnion, SetCoe.forall', restrict_iUnion_congr]
theorem restrict_sUnion_congr {S : Set (Set α)} (hc : S.Countable) :
μ.restrict (⋃₀ S) = ν.restrict (⋃₀ S) ↔ ∀ s ∈ S, μ.restrict s = ν.restrict s := by
rw [sUnion_eq_biUnion, restrict_biUnion_congr hc]
/-- This lemm... | Mathlib/MeasureTheory/Measure/Restrict.lean | 360 | 382 |
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Tactic.FieldSimp
import Mathli... | simpa [mul_comm, div_eq_mul_inv, ← map_op_mul, ← map_op_inv] using
ruzsa_triangle_inequality_div_div_div (G := Gᵐᵒᵖ) (C.map opEquiv.toEmbedding)
(B.map opEquiv.toEmbedding) (A.map opEquiv.toEmbedding)
| Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 73 | 76 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Order.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
import Batteries.WF
/-!
# Unbundled... | IsAsymm.swap _
| Mathlib/Order/RelClasses.lean | 680 | 680 |
/-
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.Logic.Encodable.Lattice
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums and product... |
section Int
section Monoid
@[to_additive HasSum.nat_add_neg_add_one]
lemma HasProd.nat_mul_neg_add_one {f : ℤ → M} (hf : HasProd f m) :
HasProd (fun n : ℕ ↦ f n * f (-(n + 1))) m := by
change HasProd (fun n : ℕ ↦ f n * f (Int.negSucc n)) m
have : Injective Int.negSucc := @Int.negSucc.inj
refine hf.hasProd_... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 328 | 343 |
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.Logic.Small.List
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.SetTheory.Ordinal.Exponen... | theorem nfp_mul_zero (a : Ordinal) : nfp (a * ·) 0 = 0 := by
rw [← Ordinal.le_zero, nfp_le_iff]
intro n
induction' n with n hn; · rfl
dsimp only; rwa [iterate_succ_apply, mul_zero]
theorem nfp_mul_eq_opow_omega0 {a b : Ordinal} (hb : 0 < b) (hba : b ≤ a ^ ω) :
nfp (a * ·) b = a ^ ω := by
rcases eq_zero_o... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 424 | 432 |
/-
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.PFunctor.Univariate.M
/-!
# Quotients of Polynomial Functors
We assume the following:
* `P`: a polynomial functor
* `W`: its W-type
* `M`: its M... | Mathlib/Data/QPF/Univariate/Basic.lean | 669 | 675 | |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... |
theorem pNilradical_eq_bot {R : Type*} [CommSemiring R] {p : ℕ} (hp : ¬ 1 < p) :
| Mathlib/FieldTheory/IsPerfectClosure.lean | 81 | 82 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | ⟨hxy₁ ∘ hyz₁, fun _ => le_trans (hxy₂ _) (hyz₂ _)⟩
lt_iff_le_not_le _ _ := Iff.rfl
le_antisymm := fun _ _ ⟨hxy₁, hxy₂⟩ ⟨hyx₁, hyx₂⟩ =>
| Mathlib/Data/Nat/PartENat.lean | 222 | 224 |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | Mathlib/Order/Height.lean | 77 | 77 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Scott Carnahan
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.MulAntidiagonal
import Mathlib... | nth_rw 1 [← zero_vadd Γ a]
exact coeff_single_smul_vadd
@[deprecated (since := "2025-01-31")] alias single_zero_smul_coeff := coeff_single_zero_smul
| Mathlib/RingTheory/HahnSeries/Multiplication.lean | 305 | 308 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | /-- The Jacobi symbol is multiplicative in its second argument. -/
theorem mul_right' (a : ℤ) {b₁ b₂ : ℕ} (hb₁ : b₁ ≠ 0) (hb₂ : b₂ ≠ 0) :
J(a | b₁ * b₂) = J(a | b₁) * J(a | b₂) := by
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 116 | 118 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,970 | 1,972 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | intro x z hz
by_cases h : x ∈ s <;> simp [h] at hz
· refine Filter.Eventually.of_forall fun x' => ?_
by_cases h' : x' ∈ s <;> simp [h', hz, hz.trans_le hy]
· filter_upwards [hs.isOpen_compl.mem_nhds h]
simp +contextual [hz]
theorem IsClosed.lowerSemicontinuousOn_indicator (hs : IsClosed s) (hy : y ≤ 0)... | Mathlib/Topology/Semicontinuous.lean | 213 | 220 |
/-
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 | 1,503 | 1,512 | |
/-
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
... | /-- `Part` extensionality -/
theorem ext' : ∀ {o p : Part α}, (o.Dom ↔ p.Dom) → (∀ h₁ h₂, o.get h₁ = p.get h₂) → o = p
| Mathlib/Data/Part.lean | 72 | 73 |
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.... | (mem_oneCocycles_iff f).1 f.2 g g⁻¹]
theorem dZero_apply_mem_oneCocycles (x : A) :
dZero A x ∈ oneCocycles A :=
congr($(dOne_comp_dZero A) x)
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 277 | 282 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | ∃ c, c ∈ { c | 0 ≤ c ∧ ∀ m, ‖f m‖ ≤ c * ∏ i, ‖m i‖ } :=
let ⟨M, hMp, hMb⟩ := f.bound
⟨M, le_of_lt hMp, hMb⟩
| Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 359 | 362 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Derivation.Killing
import Mathlib.Algebra.Lie.Killing
import Mathlib.Algebra.Lie.Sl2
import Mathlib.Algebra.Lie.Weights.Chain
import Mathlib.Line... | namespace IsKilling
variable [IsKilling R L]
/-- If the Killing form of a Lie algebra is non-singular, it remains non-singular when restricted
to a Cartan subalgebra. -/
lemma ker_restrict_eq_bot_of_isCartanSubalgebra
[IsNoetherian R L] [IsArtinian R L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
LinearM... | Mathlib/Algebra/Lie/Weights/Killing.lean | 45 | 59 |
/-
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.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,648 | 1,661 | |
/-
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
... |
/-! ### `stdSimplex` -/
| Mathlib/Analysis/Convex/Combination.lean | 461 | 463 |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | rw [rootsOfUnityEquivOfPrimitiveRoots_symm_apply, rootsOfUnity.val_mkOfPowEq_coe]
split_ifs with h
· obtain rfl := not_imp_not.mp (fun hn ↦ ne_zero_of_irreducible_X_pow_sub_C' hn H) h
rw [(pow_one _).symm.trans (root_X_pow_sub_C_pow 1 a), one_mul,
← AdjoinRoot.algebraMap_eq, AlgEquiv.commutes]... | Mathlib/FieldTheory/KummerExtension.lean | 264 | 277 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | · rw [φ.inner_map_map]
· simp
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 303 | 304 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... |
lemma toMvPolynomial_totalDegree_le (M : Matrix m n R) (i : m) :
(M.toMvPolynomial i).totalDegree ≤ 1 := by
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 99 | 101 |
/-
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... | have :
(ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
(α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
(X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv) ≫ (X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom) ≫ (X₁ ◁ (λ_ X₂).hom) := by
monoidal
rw [this]; clear this
rw [← braiding_rightUnitor]
monoidal
@[reassoc]
theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ ... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 681 | 691 |
/-
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, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... | exact ⟨∅, rfl, by convert linearIndependent_empty R M using 2 <;> aesop⟩
exact exists_finset_linearIndependent_of_le_rank
((Nat.cast_le.mpr hn).trans_eq (cast_toNat_of_lt_aleph0 (toNat_ne_zero.mp h).2))
lemma exists_linearIndependent_of_le_finrank {n : ℕ} (hn : n ≤ finrank R M) :
∃ f : Fin n → M, LinearI... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 249 | 255 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... | theorem updateCol_conjTranspose [DecidableEq m] [Star α] :
updateCol Mᴴ i (star b) = (updateRow M i b)ᴴ := by
rw [conjTranspose, conjTranspose, transpose_map, transpose_map, updateCol_transpose,
map_updateRow]
| Mathlib/Data/Matrix/RowCol.lean | 321 | 324 |
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | theorem isLindelof_singleton {x : X} : IsLindelof ({x} : Set X) := fun _ hf _ hfa ↦
⟨x, rfl, ClusterPt.of_le_nhds'
(hfa.trans <| by simpa only [principal_singleton] using pure_le_nhds x) hf⟩
| Mathlib/Topology/Compactness/Lindelof.lean | 309 | 311 |
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