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) 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... |
variable [LocallyFiniteOrderTop α]
| Mathlib/Order/Interval/Finset/Basic.lean | 724 | 726 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... |
theorem multiplicity_of_unit_right {a : α} (ha : ¬IsUnit a) (u : αˣ) : multiplicity a u = 0 :=
multiplicity_of_isUnit_right ha u.isUnit
theorem emultiplicity_le_emultiplicity_of_dvd_left {a b c : α} (hdvd : a ∣ b) :
emultiplicity b c ≤ emultiplicity a c :=
emultiplicity_le_emultiplicity_iff.2 fun n h => (pow_... | Mathlib/RingTheory/Multiplicity.lean | 549 | 574 |
/-
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 | 2,866 | 2,877 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 327 | 329 | |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | @[simps]
noncomputable def divOfHom : Multiplicative G →* AddMonoid.End k[G] where
toFun g :=
| Mathlib/Algebra/MonoidAlgebra/Division.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, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | @[simp]
theorem support_ofFinset : (ofFinset f s h h').support = ↑s ∩ Function.support f :=
| Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 168 | 169 |
/-
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 | 1,802 | 1,806 | |
/-
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.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... | ⟨⟨0, NFBelow.zero⟩⟩
theorem NFBelow.oadd {e n a b} : NF e → NFBelow a (repr e) → repr e < b → NFBelow (oadd e n a) b
| Mathlib/SetTheory/Ordinal/Notation.lean | 191 | 193 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | map_div₀ _
| Mathlib/RingTheory/Valuation/Basic.lean | 348 | 349 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | mk₃ (0 : R₁.obj' 0 ⟶ _) (R₂.map' 0 1) (R₂.map' 1 2) := homMk₃ (𝟙 _) (app' φ 0) (app' φ 1)
(app' φ 2) (by simp) (naturality' φ 0 1) (naturality' φ 1 2)
refine mono_of_epi_of_mono_of_mono' ψ (by simp)
(hR₁.exact 0).exact_toComposableArrows
(exact₂_mk _ (by simp) ?_) (by dsimp [ψ]; infer_instance) h₀ ... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 201 | 205 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | /-- A variant of `WeierstrassCurve.j_eq_zero_iff_of_char_three` without assuming a reduced ring. -/
lemma j_eq_zero_iff_of_char_three' : W.j = 0 ↔ W.b₂ ^ 6 = 0 := by
rw [j_of_char_three, Units.mul_right_eq_zero]
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 417 | 420 |
/-
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... | rintro ⟨u, hu, hus, f_an, f', huf', hf'⟩
exact ⟨u, hu, f_an, f', fun y hy => (huf' y hy).insert'.mono hus, hf'.insert.mono hus⟩
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 445 | 446 |
/-
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.MeasureTheory.MeasurableSpace.EventuallyMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.M... | calc
μ (s ∩ t) + μ (s \ t) ≤ μ (s' ∩ t) + μ (s' \ t) := by gcongr
_ = μ (s' ∩ t ∪ s' \ t) :=
(measure_union₀_aux (hs'm.inter ht) (hs'm.diff ht) <|
(@disjoint_inf_sdiff _ s' t _).aedisjoint).symm
_ = μ s' := congr_arg μ (inter_union_diff _ _)
_ = μ s := hs'
| Mathlib/MeasureTheory/Measure/NullMeasurable.lean | 262 | 269 |
/-
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... | Mathlib/Data/Finsupp/Defs.lean | 1,380 | 1,382 | |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | theorem AffineIndependent.comp_embedding {ι2 : Type*} (f : ι2 ↪ ι) {p : ι → P}
(ha : AffineIndependent k p) : AffineIndependent k (p ∘ f) := by
classical
intro fs w hw hs i0 hi0
let fs' := fs.map f
let w' i := if h : ∃ i2, f i2 = i then w h.choose else 0
have hw' : ∀ i2 : ι2, w' (f i2) = w i2 := b... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 286 | 304 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem smeval_X_pow {n : ℕ} :
(X ^ n : R[X]).smeval x = x ^ n := by
simp only [smeval_eq_sum, smul_pow, X_pow_eq_monomial, zero_smul, sum_monomial_index, one_smul]
| Mathlib/Algebra/Polynomial/Smeval.lean | 93 | 95 |
/-
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.... | group homs `G → A`. -/
@[simps] def oneCocyclesLequivOfIsTrivial [hA : A.IsTrivial] :
oneCocycles A ≃ₗ[k] Additive G →+ A where
toFun f :=
| Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 296 | 299 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Stream.Init
import Mathlib.Topology.Algebra.Semigroup
import Mathlib.Topology.StoneCech
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Hind... | Mathlib/Combinatorics/Hindman.lean | 263 | 274 | |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Ana... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 358 | 362 | |
/-
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.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | (closure_le.2 <|
Set.union_subset (Set.range_subset_iff.2 fun x => adjoin_mono Set.subset_union_left <|
| Mathlib/RingTheory/Adjoin/Basic.lean | 79 | 80 |
/-
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.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | rw [← bind_pure]; exact terminates_of_mem (mem_bind (get_mem s) (get_mem (α := β) (f (get s))))
theorem terminates_map_iff (f : α → β) (s : Computation α) : Terminates (map f s) ↔ Terminates s :=
⟨fun ⟨⟨_, h⟩⟩ =>
let ⟨_, h1, _⟩ := exists_of_mem_map h
⟨⟨_, h1⟩⟩,
@Computation.terminates_map _ _ _ _⟩
| Mathlib/Data/Seq/Computation.lean | 748 | 755 |
/-
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... | exact sdiff_le_sdiff_right h
| Mathlib/Order/Heyting/Basic.lean | 573 | 573 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,485 | 1,488 | |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | theorem finprod_false (f : False → M) : ∏ᶠ i, f i = 1 :=
finprod_of_isEmpty _
@[to_additive]
| Mathlib/Algebra/BigOperators/Finprod.lean | 198 | 201 |
/-
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 ... | @[reassoc]
lemma descOpcycles_comp {A' : C} (α : A ⟶ A') :
S.descOpcycles k hk ≫ α = S.descOpcycles (k ≫ α) (by rw [reassoc_of% hk, zero_comp]) := by
simp only [← cancel_epi S.pOpcycles, p_descOpcycles_assoc, p_descOpcycles]
/-- Via `S.pOpcycles : S.X₂ ⟶ S.opcycles`, the object `S.opcycles` identifies to the
cok... | Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 1,221 | 1,228 |
/-
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.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | ext
simp [biproduct.lift_desc]
end HasBiproduct
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 238 | 241 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 687 | 689 | |
/-
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... |
end FractionalIdeal
| Mathlib/RingTheory/DedekindDomain/Different.lean | 226 | 228 |
/-
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.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | (hd : ∀ n : ℤ, d ≠ n * n) : Function.Injective (lift r) :=
(injective_iff_map_eq_zero (lift r)).mpr fun a ha => by
have h_inj : Function.Injective ((↑) : ℤ → R) := Int.cast_injective
suffices lift r a.norm = 0 by
simp only [intCast_re, add_zero, lift_apply_apply, intCast_im, Int.cast_zero,
z... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 913 | 928 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 439 | 444 | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | (𝓝 (ENNReal.ofReal ((1 : ℝ) ^ finrank ℝ E) * μ (closedBall (0 : E) 1))) := by
refine ENNReal.Tendsto.mul ?_ (by simp) tendsto_const_nhds (by simp)
exact ENNReal.tendsto_ofReal ((tendsto_id'.2 nhdsWithin_le_nhds).pow _)
simp only [one_pow, one_mul, ENNReal.ofReal_one] at A
refine le_of_tendsto A ?_
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 486 | 490 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# The factorization axiom
In this file, we introduce a type-class `HasFactorization W₁ W₂`, which, given
two classes ... | @[reassoc (attr := simp)]
lemma fac_app {f : Arrow C} : data.i.app f ≫ data.p.app f = f.hom := by
rw [← NatTrans.comp_app, fac,Arrow.leftToRight_app]
| Mathlib/CategoryTheory/MorphismProperty/Factorization.lean | 103 | 105 |
/-
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.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | nonrec theorem IsPWO.prod {t : Set β} (hs : s.IsPWO) (ht : t.IsPWO) : IsPWO (s ×ˢ t) :=
hs.prod ht
theorem IsPWO.image_of_monotoneOn (hs : s.IsPWO) {f : α → β} (hf : MonotoneOn f s) :
IsPWO (f '' s) :=
hs.image_of_monotone_on hf
theorem IsPWO.image_of_monotone (hs : s.IsPWO) {f : α → β} (hf : Monotone f) : Is... | Mathlib/Order/WellFoundedSet.lean | 410 | 421 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... | ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact ⟨_, _, rfl, rfl⟩
| Mathlib/GroupTheory/PushoutI.lean | 88 | 93 |
/-
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.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | have := congr(inv $hU.isoSpec_inv_appTop)
rw [IsIso.inv_comp, IsIso.Iso.inv_inv, IsIso.Iso.inv_hom] at this
have := (Scheme.Γ.map_inv hU.isoSpec.inv.op).trans this
rwa [← op_inv, IsIso.Iso.inv_inv] at this
@[deprecated (since := "2024-11-16")] alias isoSpec_inv_app_top := isoSpec_inv_appTop
@[deprecated (since... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 355 | 367 |
/-
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 ... |
theorem toFilterDistortioniUnion_neBot (l : IntegrationParams) (I : Box ι) (π₀ : Prepartition I)
(hc₁ : π₀.distortion ≤ c) (hc₂ : π₀.compl.distortion ≤ c) :
(l.toFilterDistortioniUnion I c π₀).NeBot :=
((l.hasBasis_toFilterDistortion I _).inf_principal _).neBot_iff.2
| Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 492 | 496 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.Algebra.Module.LinearMap
/-!
# Continuous affine maps.
This file defines a type of bundled cont... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 274 | 274 | |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Lim... | refine ⟨fun h he ↦ ?_, fun h hin ↦ h <| (initial_iff_fiber_empty F X).mp ⟨hin⟩⟩
exact Nonempty.elim ((initial_iff_fiber_empty F X).mpr he) h
/-- An object whose fiber is inhabited is not initial. -/
lemma not_initial_of_inhabited {X : C} (x : F.obj X) (h : IsInitial X) : False :=
((initial_iff_fiber_empty F X).m... | Mathlib/CategoryTheory/Galois/Basic.lean | 226 | 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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | classical
simp_rw [lt_iff_add_one_le, le_card_iff_exists_subset_card, reduceAdd, card_eq_three,
← exists_and_left, exists_comm (α := Finset α)]
constructor
· rintro ⟨a, b, c, t, hsub, hab, hac, hbc, rfl⟩
exact ⟨a, b, c, by simp_all [insert_subset_iff]⟩
· rintro ⟨a, b, c, ha, hb, hc, hab, h... | Mathlib/Data/Finset/Card.lean | 713 | 720 |
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Analytic.CPolynomialDef
/-! # Properties of continuously polynomial functions
We expand the API a... | Mathlib/Analysis/Analytic/CPolynomial.lean | 521 | 524 | |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | variable (F : C ⥤ FintypeCat.{w}) [FiberFunctor F]
/-- Every element in the fiber of `X` lies in some connected component of `X`. -/
lemma fiber_in_connected_component (X : C) (x : F.obj X) : ∃ (Y : C) (i : Y ⟶ X) (y : F.obj Y),
F.map i y = x ∧ IsConnected Y ∧ Mono i := by
obtain ⟨ι, f, g, hl, hc, he⟩ := has_dec... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 125 | 135 |
/-
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.Order.RelClasses
import Mathlib.Data.List.Basic
/-!
# Lexicographic ordering of lists.
The lexicographic order on `List α` is defined by `L < M` iff
... | Mathlib/Data/List/Lex.lean | 235 | 238 | |
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | theorem roots_ne_zero_of_splits' {f : K[X]} (hs : Splits i f) (hf0 : natDegree (f.map i) ≠ 0) :
(f.map i).roots ≠ 0 :=
let ⟨x, hx⟩ := exists_root_of_splits' i hs fun h => hf0 <| natDegree_eq_of_degree_eq_some h
| Mathlib/Algebra/Polynomial/Splits.lean | 219 | 221 |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Eric Wieser, Bhavik Mehta,
Yaël Dillies
-/
import Mathlib.Algebra.Order.Antidiag.Pi
i... | rw [sum_of_support_subset (N := μ) f hf (fun _ x ↦ x) fun _ _ ↦ rfl]
@[simp] lemma finsuppAntidiag_empty_zero : finsuppAntidiag (∅ : Finset ι) (0 : μ) = {0} := by
ext f; simp [finsuppAntidiag, ← DFunLike.coe_fn_eq (g := f), eq_comm]
| Mathlib/Algebra/Order/Antidiag/Finsupp.lean | 50 | 53 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | namespace NormDigits
theorem digits_succ (b n m r l) (e : r + b * m = n) (hr : r < b)
| Mathlib/Data/Nat/Digits.lean | 817 | 819 |
/-
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.RightHomology
/-!
# Homology of short complexes
In this file, we shall define the homology of short complexes `S`, i.e. diagrams... | dsimp only [homologyπ]
infer_instance
lemma isIso_homologyι (hg : S.g = 0) [S.HasHomology] :
IsIso S.homologyι := by
| Mathlib/Algebra/Homology/ShortComplex/Homology.lean | 1,110 | 1,114 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | (hasFDerivAtFilter_const c x L).hasDerivAtFilter
theorem hasDerivAtFilter_zero : HasDerivAtFilter (0 : 𝕜 → F) 0 x L :=
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 651 | 653 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | rw [truncatedSup]
split_ifs with h
· obtain ⟨ℬ, hb, h⟩ := h
exact h.trans <| le_sup' id <| mem_filter.2 ⟨hb, h⟩
· exact le_top
lemma map_truncatedSup [DecidableLE β] (e : α ≃o β) (s : Finset α) (a : α) :
e (truncatedSup s a) = truncatedSup (s.map e.toEquiv.toEmbedding) (e a) := by
have : e a ∈ lowerC... | Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 132 | 141 |
/-
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
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | refine List.filter_congr ?_
intro x hx
have : x ≠ b := fun h => (nodup_cons.1 hl).1 (h ▸ hx)
simp +contextual [List.mem_filter, this, hx]
end List
theorem Option.toList_nodup : ∀ o : Option α, o.toList.Nodup
| none => List.nodup_nil
| Mathlib/Data/List/Nodup.lean | 386 | 393 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
theorem ext_getElem?_iff' {l₁ l₂ : List α} : l₁ = l₂ ↔
∀ n < max l₁.length l₂.length, l₁[n]? = l₂[n]? :=
⟨by rintro rfl _ _; rfl, ext_getElem?'⟩
@[deprecated (since := "2025-02-15")] alias ext_get?_iff' := ext_getElem?_iff'
| Mathlib/Data/List/Basic.lean | 632 | 638 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | rw [neg_Iio] at hdng
have := lhopital_zero_atTop_on_Ioi hdnf hdng
(by
intro x hx h
apply hg' _ (by rw [← neg_Iio] at hx; exact hx)
rwa [mul_comm, ← neg_eq_neg_one_mul, neg_eq_zero] at h)
(hfbot.comp tendsto_neg_atTop_atBot) (hgbot.comp tendsto_neg_atTop_atBot)
(by
simp only [mul_... | Mathlib/Analysis/Calculus/LHopital.lean | 177 | 199 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Int.Parity
im... | Mathlib/Algebra/Order/Ring/Int.lean | 100 | 100 | |
/-
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.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 140 | 144 | |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Group.Defs
/-!
# Flows and invariant sets
This file defines a flow on... | Mathlib/Dynamics/Flow.lean | 178 | 181 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | simp only [inv_eq_one_div, (Commute.one_right a).div_sub_div hab ha hb, one_mul, mul_one]
| Mathlib/Algebra/Field/Basic.lean | 129 | 129 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... |
/-- The series defining the theta function is summable if and only if `0 < im τ`. -/
lemma summable_jacobiTheta₂_term_iff (z τ : ℂ) : Summable (jacobiTheta₂_term · z τ) ↔ 0 < im τ := by
-- NB. This is a statement of no great mathematical interest; it is included largely to avoid
-- having to impose `0 < im τ` as a... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 117 | 155 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | convert Set.mem_image_of_mem (fun r' : ℝ => (x, r' • x + y)) (Set.mem_univ 0)
simp
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 804 | 805 |
/-
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... |
theorem inner_self_eq_norm_sq (x : E) : re ⟪x, x⟫ = ‖x‖ ^ 2 := by
rw [pow_two, inner_self_eq_norm_mul_norm]
theorem real_inner_self_eq_norm_mul_norm (x : F) : ⟪x, x⟫_ℝ = ‖x‖ * ‖x‖ := by
have h := @inner_self_eq_norm_mul_norm ℝ F _ _ _ x
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 355 | 360 |
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Nat
import Ma... | exact hp.dvd_of_dvd_pow h
theorem Int.Prime.dvd_pow' {n : ℤ} {k p : ℕ} (hp : Nat.Prime p) (h : (p : ℤ) ∣ n ^ k) :
| Mathlib/RingTheory/Int/Basic.lean | 88 | 90 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | refine Nat.eq_of_mul_eq_mul_left (by decide)
(Nat.add_left_cancel (Eq.trans ?_ (Nat.mod_add_div n 2).symm))
rw [mod_two_of_bodd, bodd_add_div2]
| Mathlib/Data/Nat/Bits.lean | 113 | 116 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... | rw [h₁ ⟨a, ha₁.1, ha₂.1⟩, h₂ ⟨a, ha₁.2, ha₂.2⟩]
/-- An intersection of blocks is a block. -/
@[to_additive "An intersection of blocks is a block."]
lemma IsBlock.iInter {ι : Sort*} {B : ι → Set X} (hB : ∀ i, IsBlock G (B i)) :
IsBlock G (⋂ i, B i) := by
| Mathlib/GroupTheory/GroupAction/Blocks.lean | 314 | 319 |
/-
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.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | change (rightShiftAddEquiv K L n a n' hn').symm 0 = 0
apply map_zero
@[simp]
lemma leftShift_zero (a n' : ℤ) (hn' : n + a = n') :
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 208 | 212 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | rcases m with m | m <;> rcases n with n | n <;> try {rfl}
<;> simp only [bitwise, natBitwise, Bool.not_false, Bool.or_true,
cond_false, cond_true, lor, Nat.ldiff, Bool.and_true, negSucc.injEq,
| Mathlib/Data/Int/Bitwise.lean | 251 | 253 |
/-
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... |
variable (G) in
lemma ZModModule.two_le_char [NeZero n] [Nontrivial G] : 2 ≤ n := by
have := NeZero.ne n
have := char_ne_one n G
omega
lemma ZModModule.periodicPts_add_left [NeZero n] (x : G) : periodicPts (x + ·) = .univ :=
Set.eq_univ_of_forall fun y ↦ ⟨n, NeZero.pos n, by
simpa [char_nsmul_eq_zero, IsP... | Mathlib/Data/ZMod/Basic.lean | 1,197 | 1,209 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 680 | 683 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... |
@[symm]
theorem symm (h : v₁.IsEquiv v₂) : v₂.IsEquiv v₁ := fun _ _ => Iff.symm (h _ _)
@[trans]
theorem trans (h₁₂ : v₁.IsEquiv v₂) (h₂₃ : v₂.IsEquiv v₃) : v₁.IsEquiv v₃ := fun _ _ =>
Iff.trans (h₁₂ _ _) (h₂₃ _ _)
theorem of_eq {v' : Valuation R Γ₀} (h : v = v') : v.IsEquiv v' := by subst h; rfl
theorem map {v' ... | Mathlib/RingTheory/Valuation/Basic.lean | 421 | 438 |
/-
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.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | @funMap (L₁.sum L₂) S _ n (Sum.inl f) = funMap f :=
rfl
@[simp]
| Mathlib/ModelTheory/Basic.lean | 746 | 749 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | (h : ∀ s : ∀ i, Set (α i), (∀ i, MeasurableSet (s i)) → μ' (pi univ s) = ∏ i, μ i (s i)) :
Measure.pi μ = μ' :=
pi_eq_generateFrom (fun _ => generateFrom_measurableSet) (fun _ => isPiSystem_measurableSet)
(fun i => (μ i).toFiniteSpanningSetsIn) h
variable (μ)
| Mathlib/MeasureTheory/Constructions/Pi.lean | 278 | 284 |
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-! # Power funct... |
lemma cpow_ofNat_mul (x : ℂ) (n : ℕ) [n.AtLeastTwo] (y : ℂ) :
x ^ (ofNat(n) * y) = (x ^ y) ^ ofNat(n) :=
cpow_nat_mul x n y
lemma cpow_mul_nat (x y : ℂ) (n : ℕ) : x ^ (y * n) = (x ^ y) ^ n := by
rw [mul_comm, cpow_nat_mul]
| Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | 114 | 120 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Rat.Cast.CharZero
import Mathlib.Tactic.Positivity.Core
/-!
# Casts of rational numbers in... | @[mono]
theorem cast_strictMono : StrictMono ((↑) : ℚ → K) := fun p q => by
simpa only [sub_pos, cast_sub] using cast_pos_of_pos (K := K) (q := q - p)
| Mathlib/Data/Rat/Cast/Order.lean | 31 | 33 |
/-
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, Kim Morrison
-/
import Mathlib.Data.List.Chain
/-!
# Ranges of naturals as lists
This file shows basic results about `List.iota`, `List.range`, `List.range... | Mathlib/Data/List/Range.lean | 203 | 205 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | refine ⟨f x, fun a ↦ ?_⟩
obtain ⟨a, rfl⟩ := hf a
obtain ⟨n, rfl⟩ := hx a
exact ⟨n, (map_zpow _ _ _).symm⟩
@[to_additive]
theorem orderOf_eq_card_of_forall_mem_zpowers {g : α} (hx : ∀ x, x ∈ zpowers g) :
orderOf g = Nat.card α := by
rw [← Nat.card_zpowers, (zpowers g).eq_top_iff'.mpr hx, card_top]
@[depr... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 213 | 231 |
/-
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, Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndependent.Lemmas
import Mathlib.LinearAlgebra.L... | def sumExtendIndex (hs : LinearIndependent K v) : Set V :=
LinearIndepOn.extend hs.linearIndepOn_id (subset_univ _) \ range v
| Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 67 | 69 |
/-
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.... | exact StrongHomClass.map_rel g _ _
| imp _ _ ih1 ih2 => rw [BoundedFormula.Realize, ih1, ih2, BoundedFormula.Realize]
| all _ ih3 =>
rw [BoundedFormula.Realize, BoundedFormula.Realize]
constructor
· intro h a
| Mathlib/ModelTheory/Semantics.lean | 888 | 893 |
/-
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.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
@[fun_prop]
theorem Continuous.finInit {f : X → ∀ j : Fin (n + 1), π j} (hf : Continuous f) :
| Mathlib/Topology/Constructions.lean | 833 | 835 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Subsemigroup.Membership
import Mathlib.Algebra.Group... | (NonUnitalSubsemiring.gi R).gc.l_bot
@[simp]
theorem closure_univ : closure (Set.univ : Set R) = ⊤ :=
@coe_top R _ ▸ closure_eq ⊤
theorem closure_union (s t : Set R) : closure (s ∪ t) = closure s ⊔ closure t :=
(NonUnitalSubsemiring.gi R).gc.l_sup
theorem closure_iUnion {ι} (s : ι → Set R) : closure (⋃ i, s i)... | Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | 500 | 510 |
/-
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.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | Mathlib/Algebra/RingQuot.lean | 649 | 652 | |
/-
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.NonUnitalHom
import Mathlib.LinearAlgebra.TensorProduct.Basic
/-!
# Facts about algebras involving bilinear maps and tensor pro... |
/-- The multiplication in a non-unital algebra is a bilinear map.
| Mathlib/Algebra/Algebra/Bilinear.lean | 140 | 142 |
/-
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.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | let ⟨⟨a, ma⟩⟩ := l.2 ⟨⟨_, mb⟩⟩
⟨a, ma, r ma mb⟩⟩⟩
theorem liftRel_bind {δ} (R : α → β → Prop) (S : γ → δ → Prop) {s1 : Computation α}
{s2 : Computation β} {f1 : α → Computation γ} {f2 : β → Computation δ} (h1 : LiftRel R s1 s2)
| Mathlib/Data/Seq/Computation.lean | 950 | 954 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | dsimp
simp only [map_v, comp_v _ _ h p _ q rfl (by omega), Φ.map_comp]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 813 | 814 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 559 | 562 | |
/-
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.Pretriangulated
import Mathlib.CategoryTheory.Triangulated.Triangulated
import Mathlib.CategoryTheory.ComposableArrows
/-! The... | refine Triangle.isoMk _ _ (Iso.refl _) (Iso.refl _) (isoOfHomotopyEquiv
(mappingConeCompHomotopyEquiv f g)) (by aesop_cat) (by simp) ?_
dsimp [mappingConeCompTriangleh]
rw [CategoryTheory.Functor.map_id, comp_id, ← Functor.map_comp_assoc]
congr 2
exact (mappingConeCompHomotopyEquiv_comm₂ f g).symm
noncom... | Mathlib/Algebra/Homology/HomotopyCategory/Triangulated.lean | 179 | 188 |
/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | (hp₂ : p₂ ∈ s) {t : R} (ht : t ≤ 0) : s.WOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x := by
refine ⟨p₂, hp₂, p₁, hp₁, ?_⟩
rw [vadd_vsub, ← neg_neg t, neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]
| Mathlib/Analysis/Convex/Side.lean | 294 | 296 |
/-
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.LinearAlgebra.FreeModule.Basic
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.Probability.ConditionalProbability
/-!
# s-fin... | instance [SFinite μ] : IsFiniteMeasure μ.toFinite := by
rw [Measure.toFinite]
infer_instance
instance [SFinite μ] [NeZero μ] : IsProbabilityMeasure μ.toFinite := by
apply ProbabilityTheory.cond_isProbabilityMeasure
simp [ne_eq, ← compl_mem_ae_iff, ae_toFiniteAux]
lemma absolutelyContinuous_toFinite (μ : Measu... | Mathlib/MeasureTheory/Measure/WithDensityFinite.lean | 93 | 105 |
/-
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.Algebra.Group.Indicator
import Mathlib.Data.Int.Cast.Pi
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.MeasureTheory.MeasurableSpace... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 818 | 829 | |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.Algebra.Group.Fin.Basic
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Tactic.Abel
/-!
# Circulant matrices
This file contains the defini... | simp only [mul_apply, circulant_apply, mul_comm]
refine Fintype.sum_equiv ((Equiv.subLeft i).trans (Equiv.addRight j)) _ _ ?_
intro x
simp only [Equiv.trans_apply, Equiv.subLeft_apply, Equiv.coe_addRight, add_sub_cancel_right,
mul_comm]
congr 2
abel
| Mathlib/LinearAlgebra/Matrix/Circulant.lean | 127 | 133 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
theorem setLAverage_lt_top : ∫⁻ x in s, f x ∂μ ≠ ∞ → ⨍⁻ x in s, f x ∂μ < ∞ :=
| Mathlib/MeasureTheory/Integral/Average.lean | 153 | 155 |
/-
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.Order.Archimedean.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib... | @[to_additive Summable.tsum_pos]
protected theorem Multipliable.one_lt_tprod (hsum : Multipliable g) (hg : ∀ i, 1 ≤ g i) (i : ι)
| Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 249 | 250 |
/-
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
... | lemma Finset.centerMass_smul_left {c : R'} [Module R' R] [Module R' E] [SMulCommClass R' R R]
[IsScalarTower R' R R] [SMulCommClass R R' E] [IsScalarTower R' R E] (hc : c ≠ 0) :
| Mathlib/Analysis/Convex/Combination.lean | 70 | 71 |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.Field.ZMod
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.LocalRing.ResidueField.... | norm_cast
rw [← padicNorm.dvd_iff_norm_le]
exact mod_cast pow_dvd_nthHom_sub f_compat r k j hj
/-- `nthHomSeq f_compat r` bundles `PadicInt.nthHom f r`
as a Cauchy sequence of rationals with respect to the `p`-adic norm.
The `n`th value of the sequence is `((f n r).val : ℚ)`.
| Mathlib/NumberTheory/Padics/RingHoms.lean | 505 | 511 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Peter Nelson
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determina... | convert hdet <;>
simp [← Matrix.ext_iff, projVandermonde_apply, u]
/-- The formula for the determinant of a Vandermonde matrix. -/
theorem det_vandermonde (v : Fin n → R) :
det (vandermonde v) = ∏ i : Fin n, ∏ j ∈ Ioi i, (v j - v i) := by
| Mathlib/LinearAlgebra/Vandermonde.lean | 205 | 210 |
/-
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.Algebra.Order.Group.Nat
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Logic.Enco... | cases n with
| zero => simp only [Function.iterate_zero, id]
| succ n =>
| Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 73 | 75 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
/-!
# Extensionality lemmas for monoid and group structures
In this file we prove extensionality lemmas... | @[to_additive (attr := ext)]
theorem CommMonoid.ext {M : Type*} ⦃m₁ m₂ : CommMonoid M⦄
(h_mul : (letI := m₁; HMul.hMul : M → M → M) = (letI := m₂; HMul.hMul : M → M → M)) : m₁ = m₂ :=
CommMonoid.toMonoid_injective <| Monoid.ext h_mul
| Mathlib/Algebra/Group/Ext.lean | 56 | 59 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... |
theorem lt_irrefl {f : CauSeq α abs} : ¬f < f
| h => not_limZero_of_pos h (by simp [zero_limZero])
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 645 | 647 |
/-
Copyright (c) 2018 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Data.Nat.Find
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Basic properties of holors
Holors... | end Holor
| Mathlib/Data/Holor.lean | 330 | 350 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import... | ordConnected_Ico.isPreconnected
theorem isConnected_Ici : IsConnected (Ici a) :=
⟨nonempty_Ici, isPreconnected_Ici⟩
theorem isConnected_Iic : IsConnected (Iic a) :=
⟨nonempty_Iic, isPreconnected_Iic⟩
theorem isConnected_Ioi [NoMaxOrder α] : IsConnected (Ioi a) :=
⟨nonempty_Ioi, isPreconnected_Ioi⟩
| Mathlib/Topology/Order/IntermediateValue.lean | 401 | 410 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... |
theorem eqOn_comp_right_iff : s.EqOn (g₁ ∘ f) (g₂ ∘ f) ↔ (f '' s).EqOn g₁ g₂ :=
| Mathlib/Data/Set/Function.lean | 554 | 555 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | Mathlib/NumberTheory/Padics/PadicIntegers.lean | 599 | 601 | |
/-
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.Order.Filter.Tendsto
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Data.Finset.Lat... | hf.bicomp_mono (fun _x _x' hx _y _y' hy => max_le_max hx hy) hg
| Mathlib/Order/Filter/Extr.lean | 507 | 508 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | (hg : ContDiffOn 𝕜 n g s) : ContDiffOn 𝕜 n (fun x : E => (f x, g x)) s := fun x hx =>
(hf x hx).prodMk (hg x hx)
@[deprecated (since := "2025-03-09")]
alias ContDiffOn.prod := ContDiffOn.prodMk
/-- The cartesian product of `C^n` functions at a point is `C^n`. -/
theorem ContDiffAt.prodMk {f : E → F} {g : E → ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 516 | 526 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.