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) 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... | theorem exists_Icc_mem_subset_of_mem_nhdsLE [OrderTopology α] {a : α} {s : Set α}
(hs : s ∈ 𝓝[≤] a) : ∃ b ≤ a, Icc b a ∈ 𝓝[≤] a ∧ Icc b a ⊆ s := by
simpa only [Icc_toDual, toDual.surjective.exists] using
exists_Icc_mem_subset_of_mem_nhdsGE (α := αᵒᵈ) (a := toDual a) hs
| Mathlib/Topology/Order/Basic.lean | 407 | 410 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence... | codisjoint_iff.2 mrange_inl_sup_mrange_inr
@[to_additive] theorem mrange_eq (f : M ∗ N →* P) :
MonoidHom.mrange f = MonoidHom.mrange (f.comp inl) ⊔ MonoidHom.mrange (f.comp inr) := by
| Mathlib/GroupTheory/Coprod/Basic.lean | 263 | 266 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... |
theorem tendsto_translationNumber_aux : Tendsto f.transnumAuxSeq atTop (𝓝 <| τ f) :=
(cauchySeq_of_le_geometric_two fun n => le_of_lt <| f.transnumAuxSeq_dist_lt n).tendsto_limUnder
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 587 | 590 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,373 | 2,375 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | comm₁₂ := by
dsimp
rw [Category.comp_id, ← L.map_comp, kernel.lift_ι] }
apply (ShortComplex.exact_iff_of_epi_of_isIso_of_mono φ).2
apply ShortComplex.exact_of_f_is_kernel
| Mathlib/CategoryTheory/Abelian/Exact.lean | 295 | 299 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | have cpos : 0 < ‖c‖ := by positivity
have cnpos : 0 < ‖c^(n+1)‖ := by rw [norm_zpow]; exact xεpos.trans hn.2
refine ⟨-(n+1), ?_, ?_, ?_, ?_⟩
· show c ^ (-(n + 1)) ≠ 0; exact zpow_ne_zero _ (norm_pos_iff.1 cpos)
· show p ((c ^ (-(n + 1))) • x) < ε
| Mathlib/Analysis/Seminorm.lean | 1,196 | 1,200 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.Topology.MetricSpace.ThickenedIndicator
/-!
# Spaces where indicators of closed sets have ... | μ F = ν F := by
have ν_finite : IsFiniteMeasure ν := by
constructor
have whole := h 1
simp only [BoundedContinuousFunction.coe_one, Pi.one_apply, ENNReal.coe_one, lintegral_const,
one_mul] at whole
simp [← whole]
have obs_μ := HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed μ
hav... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 207 | 221 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | @[to_additive "For any `x : M ⧸ S` and any `0 < ε`, there is `m : M` such that `mk' S m = x`
and `‖m‖ < ‖x‖ + ε`."]
lemma exists_norm_mk_lt (x : M ⧸ S) (hε : 0 < ε) : ∃ m : M, m = x ∧ ‖m‖ < ‖x‖ + ε :=
norm_lt_iff.1 <| lt_add_of_pos_right _ hε
| Mathlib/Analysis/Normed/Group/Quotient.lean | 186 | 189 |
/-
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.Action.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Scalar
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Group.Submon... |
@[to_additive]
| Mathlib/GroupTheory/GroupAction/Defs.lean | 70 | 71 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,240 | 1,243 | |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | · filter_upwards [self_mem_nhdsWithin] with δ δ_pos
simp only [closure_subset_thickening δ_pos E x_mem_closure, mulIndicator_of_mem, x_mem_closure]
· have obs := eventually_not_mem_thickening_of_infEdist_pos x_mem_closure
filter_upwards [mem_nhdsWithin_of_mem_nhds obs, self_mem_nhdsWithin]
with δ x_no... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 246 | 257 |
/-
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.... |
section lim
| Mathlib/Topology/Basic.lean | 197 | 198 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,212 | 1,216 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | refine ⟨⟨j - c.sizeUpTo i, ?_⟩, ?_⟩
· rw [tsub_lt_iff_left, ← sizeUpTo_succ']
· exact h.2
· exact h.1
· rw [Fin.ext_iff]
exact add_tsub_cancel_of_le h.1
/-- The embeddings of different blocks of a composition are disjoint. -/
theorem disjoint_range {i₁ i₂ : Fin c.length} (h : i₁ ≠ i₂) :
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 362 | 378 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... |
open LaurentPolynomial hiding C
| Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 292 | 293 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | intro hind h hcop
rw [hind h (Nat.Coprime.coprime_mul_left hcop)]; clear hind
replace hprime := hprime.nat_prime
have hdiv := (Nat.Prime.coprime_iff_not_dvd hprime).1 (Nat.Coprime.coprime_mul_right hcop)
haveI := Fact.mk hprime
rw [minpoly_eq_pow (h.pow_of_coprime a (Nat.Coprime.coprime_mul_left... | Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 175 | 193 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | ext i a b
dsimp
rw [gradedCommAux_lof_tmul, LinearMap.map_smul_of_tower, gradedCommAux_lof_tmul, smul_smul,
mul_comm i.2 i.1, Int.units_mul_self, one_smul]
/-- The braiding operation for tensor products of externally `ι`-graded algebras.
| Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 93 | 98 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,522 | 1,525 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
end reflection
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 730 | 731 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | · apply lt_of_lt_of_le hx <| Nat.pow_le_pow_right (le_succ _) (le_add_right _ _)
| Mathlib/Data/Nat/Bitwise.lean | 376 | 376 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.HurwitzZetaEven
import Mathlib.NumberTheory.LSeries.HurwitzZetaOdd
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
/-!
# The... | /-- Formula for `hurwitzZeta s` as a Dirichlet series in the convergence range. We
restrict to `a ∈ Icc 0 1` to simplify the statement. -/
lemma hasSum_hurwitzZeta_of_one_lt_re {a : ℝ} (ha : a ∈ Icc 0 1) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℕ ↦ 1 / (n + a : ℂ) ^ s) (hurwitzZeta a s) := by
convert (hasSum_nat... | Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | 67 | 75 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 236 | 237 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.Pseudo.Basic
import Mathlib.Topology.MetricSpace.Pseudo.Le... | SecondCountableTopology α := by
refine secondCountable_of_almost_dense_set fun ε ε0 => ?_
rcases H ε ε0 with ⟨β, fβ, F, hF⟩
let Finv := rangeSplitting F
| Mathlib/Topology/MetricSpace/Basic.lean | 165 | 168 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | def any (f : ∀ x, β x → Bool) (s : Finmap β) : Bool :=
s.foldl (fun x y z => x || f y z)
(fun _ _ _ _ => by simp_rw [Bool.or_assoc, Bool.or_comm, imp_true_iff]) false
/-- `all f s` returns `true` iff `f v = true` for all values `v` in `s`. -/
| Mathlib/Data/Finmap.lean | 353 | 357 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Subobject.MonoOver
/-!
# Su... | have : (Limits.prod.fst : A ⨯ A ⟶ _) = Limits.prod.snd := by simp [← cancel_epi (diag A)]
rw [← prod.lift_fst f g, this, prod.lift_snd]
/-- If `A` is subterminal, it is isomorphic to `A ⨯ A`. -/
@[simps!]
| Mathlib/CategoryTheory/Subterminal.lean | 97 | 101 |
/-
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.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | Mathlib/Data/Finset/Image.lean | 783 | 786 | |
/-
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.Pow.Real
/-!
# Power function... | theorem rpow_lt_one {x : ℝ≥0} {z : ℝ} (hx1 : x < 1) (hz : 0 < z) : x ^ z < 1 :=
Real.rpow_lt_one (coe_nonneg x) hx1 hz
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 337 | 338 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | obtain rfl | hr := hr.eq_or_lt
· rw [logb_zero, Int.log_zero_right, Int.floor_zero]
| Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 357 | 358 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Junyan Xu
-/
import Mathlib.Data.Fintype.Order
import Mathlib.FieldTheory.IntermediateField.Adjoin.Basic
/-!
# Extension of field embeddings
`IntermediateField.exis... | rw [eq_iff_le_carrier_eq, le_iff]
exact ⟨fun h ↦ ⟨h.2, h.1.2⟩, fun h ↦ ⟨⟨h.1.le, h.2⟩, h.1⟩⟩
theorem lt_iff_le_carrier_ne : L₁ < L₂ ↔ L₁ ≤ L₂ ∧ L₁.carrier ≠ L₂.carrier := by
rw [lt_iff_le_and_ne, and_congr_right]; intro h; simp_rw [Ne, eq_iff_le_carrier_eq, h, true_and]
theorem lt_iff : L₁ < L₂ ↔
∃ h : L₁.c... | Mathlib/FieldTheory/Extension.lean | 75 | 88 |
/-
Copyright (c) 2021 François Sunatori. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: François Sunatori
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic
/-!
# Isometries o... | · rw [← I_re]
exact @LinearIsometry.re_apply_eq_re f.toLinearIsometry h I
· apply @LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re f.toLinearIsometry
intro z
| Mathlib/Analysis/Complex/Isometry.lean | 119 | 122 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 352 | 353 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... | theorem associated_generator_span_self [IsPrincipalIdealRing R] [IsDomain R] (r : R) :
Associated (generator <| Ideal.span {r}) r := by
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 129 | 130 |
/-
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.Data.NNReal.Basic
import Mathlib.Order.Fin.Tuple
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.... |
/-- The interior of a box. -/
| Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 390 | 392 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | namespace GradedObject
variable {C : Type u} [Category.{v} C]
section
| Mathlib/CategoryTheory/GradedObject.lean | 144 | 148 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.L... | theorem LinearEquiv.det_mul_det_symm {A : Type*} [CommRing A] [Module A M] (f : M ≃ₗ[A] M) :
LinearMap.det (f : M →ₗ[A] M) * LinearMap.det (f.symm : M →ₗ[A] M) = 1 := by
simp [← LinearMap.det_comp]
| Mathlib/LinearAlgebra/Determinant.lean | 411 | 413 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | Mathlib/Algebra/Ring/Commute.lean | 304 | 308 | |
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | (hy : y ∈ (extChartAt (IB.prod 𝓘(𝕜, F)) x).target) :
writtenInExtChartAt (IB.prod 𝓘(𝕜, F)) (IB.prod 𝓘(𝕜, F)) x
(trivializationAt F E x.proj) y = y :=
writtenInExtChartAt_chartAt_comp _ hy
theorem FiberBundle.writtenInExtChartAt_trivializationAt_symm {x : TotalSpace F E} {y}
| Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 148 | 153 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 1,072 | 1,074 | |
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Bilinear form
This file defines a bilinear form over a module. ... |
variable (B D B₁ D₁)
| Mathlib/LinearAlgebra/BilinearForm/Basic.lean | 103 | 104 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | theorem Convex.interior_closure_eq_interior_of_nonempty_interior {s : Set E} (hs : Convex 𝕜 s)
(hs' : (interior s).Nonempty) : interior (closure s) = interior s := by
refine subset_antisymm ?_ (interior_mono subset_closure)
intro y hy
rcases hs' with ⟨x, hx⟩
have h := AffineMap.lineMap_apply_one (k := 𝕜) ... | Mathlib/Analysis/Convex/Topology.lean | 296 | 301 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | A ^ (n - 1) = A ^ (n - 1) * A * A⁻¹ := by
rw [mul_assoc, mul_nonsing_inv _ h, mul_one]
| Mathlib/LinearAlgebra/Matrix/ZPow.lean | 145 | 146 |
/-
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... | theorem ContDiffOn.congr_mono (hf : ContDiffOn 𝕜 n f s) (h₁ : ∀ x ∈ s₁, f₁ x = f x) (hs : s₁ ⊆ s) :
ContDiffOn 𝕜 n f₁ s₁ :=
(hf.mono hs).congr h₁
/-- If a function is `C^n` on a set with `n ≥ 1`, then it is differentiable there. -/
theorem ContDiffOn.differentiableOn (h : ContDiffOn 𝕜 n f s) (hn : 1 ≤ n) :
... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 559 | 605 |
/-
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.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | refine (((ConvexOn.add ?_ ?_).add ?_).add_const _).congr log_doublingGamma_eq.symm
· convert
convexOn_log_Gamma.comp_affineMap (DistribMulAction.toLinearMap ℝ ℝ (1 / 2 : ℝ)).toAffineMap
using 1
· simpa only [zero_div] using (preimage_const_mul_Ioi (0 : ℝ) one_half_pos).symm
· ext1 x
simp o... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 395 | 415 |
/-
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, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | `fun x : α ↦ f x * g x` also tends to zero on `l`. -/
theorem Tendsto.tendsto_mul_zero_of_disjoint_cocompact_left {f g : α → M} {l : Filter α}
(hf : Disjoint (map f l) (cocompact M)) (hg : Tendsto g l (𝓝 0)):
Tendsto (fun x ↦ f x * g x) l (𝓝 0) :=
tendsto_mul_prod_nhds_zero_of_disjoint_cocompact hf |>.comp ... | Mathlib/Topology/Algebra/Monoid.lean | 438 | 442 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 1,406 | 1,408 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | · simp [sub_eq_add_neg]
· lift z to ℝ using him
simpa using hre.ne
theorem continuousWithinAt_log_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0) (him : z.im = 0) :
ContinuousWithinAt log { z : ℂ | 0 ≤ z.im } z := by
convert
(continuous_ofReal.continuousAt.comp_continuousWithinAt
(continuou... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 177 | 189 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Data.ENNReal.Action
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.MetricSpace.Algebra
... | @[bound]
theorem norm_smul_le (r : α) (x : β) : ‖r • x‖ ≤ ‖r‖ * ‖x‖ := by
| Mathlib/Analysis/Normed/MulAction.lean | 29 | 30 |
/-
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, Mario Carneiro
-/
import Mathlib.Data.Countable.Defs
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Nat.Prime.Infinite
import Mathlib.Data... |
/-- The only prime divisor of positive prime power `p^k` is `p` itself -/
lemma primeFactors_prime_pow (hk : k ≠ 0) (hp : Prime p) :
(p ^ k).primeFactors = {p} := by simp [primeFactors_pow p hk, hp]
| Mathlib/Data/Nat/PrimeFin.lean | 114 | 117 |
/-
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.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | cases nonempty_fintype α
obtain ⟨l, hl⟩ := (truncSwapFactors f).out
induction l generalizing f with
| nil =>
simp only [one, hl.left.symm, List.prod_nil, forall_true_iff]
| cons g l ih =>
rcases hl.2 g (by simp) with ⟨x, y, hxy⟩
rw [← hl.1, List.prod_cons, hxy.2]
exact swap_mul _ _ _ hxy.1 (ih... | Mathlib/GroupTheory/Perm/Sign.lean | 100 | 111 |
/-
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.... | variable {𝒜}
/-- The vanishing ideal of a set `t` of points of the projective spectrum of a commutative ring `R`
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 81 | 83 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Lookmap
import Mathlib.Data.Si... | theorem dlookup_append (l₁ l₂ : List (Sigma β)) (a : α) :
(l₁ ++ l₂).dlookup a = (l₁.dlookup a).or (l₂.dlookup a) := by
induction l₁ with
| nil => rfl
| cons x l₁ IH =>
| Mathlib/Data/List/Sigma.lean | 311 | 315 |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | lemma schnirelmannDensity_setOf_even : schnirelmannDensity (setOf Even) = 0 :=
schnirelmannDensity_eq_zero_of_one_not_mem <| by simp
| Mathlib/Combinatorics/Schnirelmann.lean | 209 | 210 |
/-
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.Bilinear
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Group.Pointwise.Finset.Basic
import Mathlib.Algebra.Group.Pointwise.Set.B... | mul_zero := mul_bot
left_distrib := mul_sup
right_distrib := sup_mul
instance : Pow (Submodule R A) ℕ where
| Mathlib/Algebra/Algebra/Operations.lean | 291 | 295 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib... | simpa only [← LinearMap.ker_eq_bot, ker_toLin'_eq_bot_iff, not_imp_not] using h
have :
M *
LinearMap.toMatrix'
((LinearEquiv.ofInjectiveEndo (Matrix.toLin' M) this).symm : (n → K) →ₗ[K] n → K) =
1 := by
refine Matrix.toLin'.injective (LinearMap.ext fun v => ?_)
... | Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 114 | 132 |
/-
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.Algebra
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.LinearAlgebra.Dual.Lemmas
import Mathlib.RingTheory.Algebra... | /-- Auxiliary construction to a splitting field of a polynomial, which removes
`n` (arbitrarily-chosen) factors.
| Mathlib/FieldTheory/SplittingField/Construction.lean | 103 | 104 |
/-
Copyright (c) 2022 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.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Group.Measure
/-!
# Bochner Integration on Groups
We develop properties of inte... | theorem integrable_comp_div_left (f : G → F) [IsInvInvariant μ] [IsMulLeftInvariant μ] (g : G) :
Integrable (fun t => f (g / t)) μ ↔ Integrable f μ := by
refine ⟨fun h => ?_, fun h => h.comp_div_left g⟩
convert h.comp_inv.comp_mul_left g⁻¹
simp_rw [div_inv_eq_mul, mul_inv_cancel_left]
| Mathlib/MeasureTheory/Group/Integral.lean | 150 | 154 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | private theorem ghostFun_intCast (i : ℤ) : ghostFun (i : 𝕎 R) = i :=
show ghostFun i.castDef = _ by
| Mathlib/RingTheory/WittVector/Basic.lean | 187 | 188 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
/-!
# The category of arrows
The category of arrows, with morphisms commutative squares.
We set this up as a specialization ... |
instance {X Y : T} : CoeOut (X ⟶ Y) (Arrow T) where
coe := mk
| Mathlib/CategoryTheory/Comma/Arrow.lean | 91 | 94 |
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | /-- The sine of an angle in a right-angled triangle multiplied by the hypotenuse equals the
opposite side. -/
theorem sin_angle_mul_dist_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
Real.sin (∠ p₂ p₃ p₁) * dist p₁ p₃ = dist p₁ p₂ := by
rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_i... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 417 | 423 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | f =ᵐ[μ] 0 := by
let s := denseSeq E
have hs : DenseRange s := denseRange_denseSeq E
have hf' : ∀ᵐ x ∂μ, ∀ n : ℕ, inner (s n) (f x) = (0 : 𝕜) := ae_all_iff.mpr fun n => hf (s n)
refine hf'.mono fun x hx => ?_
rw [Pi.zero_apply, ← @inner_self_eq_zero 𝕜]
have h_closed : IsClosed {c : E | inner c (f x) = ... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 58 | 68 |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... | theorem comp₂Measurable_toGerm [PseudoMetrizableSpace β] [MeasurableSpace β] [BorelSpace β]
[PseudoMetrizableSpace γ] [SecondCountableTopologyEither β γ]
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 450 | 451 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | rw [← ncard_inter_add_ncard_diff_eq_ncard s t hs, ← ncard_inter_add_ncard_diff_eq_ncard t s ht,
inter_comm, add_lt_add_iff_left]
theorem ncard_add_ncard_compl (s : Set α) (hs : s.Finite := by toFinite_tac)
| Mathlib/Data/Set/Card.lean | 928 | 931 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 572 | 580 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | theorem inseparable_one_iff_norm {a : E} : Inseparable a 1 ↔ ‖a‖ = 0 := by
rw [Metric.inseparable_iff, dist_one_right]
@[to_additive]
lemma dist_one_left (a : E) : dist 1 a = ‖a‖ := by rw [dist_comm, dist_one_right]
| Mathlib/Analysis/Normed/Group/Basic.lean | 407 | 412 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | /-- If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
all points. -/
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 573 | 574 |
/-
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... | theorem restrict_inter_add_diff₀ (s : Set α) (ht : NullMeasurableSet t μ) :
μ.restrict (s ∩ t) + μ.restrict (s \ t) = μ.restrict s := by
| Mathlib/MeasureTheory/Measure/Restrict.lean | 221 | 222 |
/-
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.Order.Group.Indicator
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# Operations on outer measures
In this file we defi... |
@[simp]
theorem restrict_empty (m : OuterMeasure α) : restrict ∅ m = 0 :=
| Mathlib/MeasureTheory/OuterMeasure/Operations.lean | 306 | 308 |
/-
Copyright (c) 2022 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.FiberBundle.Basic
/-!
# Standard constructions on fiber bundles
This file contains... | rw [(Prod.isInducing_diag F₁ E₁ F₂ E₂).continuousOn_iff]
have H₁ : Continuous fun p : B × F₁ × F₂ ↦ ((p.1, p.2.1), (p.1, p.2.2)) := by fun_prop
refine (e₁.continuousOn_symm.prodMap e₂.continuousOn_symm).comp H₁.continuousOn ?_
exact fun x h ↦ ⟨⟨h.1.1, mem_univ _⟩, ⟨h.1.2, mem_univ _⟩⟩
variable (e₁ e₂)
| Mathlib/Topology/FiberBundle/Constructions.lean | 180 | 185 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | have eq₂ : ((Γ₀.splitting (N₂.obj P).X).cofan _).inj (Splitting.IndexSet.id (op ⦋n⦌)) ≫
(N₂.map (Γ₂N₂.natTrans.app P)).f.f n = PInfty.f n ≫ P.p.app (op ⦋n⦌) := by
dsimp
rw [PInfty_on_Γ₀_splitting_summand_eq_self_assoc, Γ₂N₂.natTrans_app_f_app]
dsimp
rw [Γ₂N₂ToKaroubiIso_hom_app, assoc, Splitting... | Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 202 | 207 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | rintro ⟨n, n_gt_0, eq'⟩
exact ⟨n, n_gt_0, by rw [← isConj_one_right, ← eq']; exact h.pow n⟩
/-- The image of an element of finite order has finite order. -/
| Mathlib/GroupTheory/OrderOfElement.lean | 126 | 129 |
/-
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.OuterMeasure.OfFunction
import Mathlib.MeasureTheory.PiSystem
/-!
# The Caratheodory σ-algebra of an outer measure
Give... | theorem isCaratheodory_compl_iff : IsCaratheodory m sᶜ ↔ IsCaratheodory m s :=
⟨fun h => by simpa using isCaratheodory_compl m h, isCaratheodory_compl m⟩
| Mathlib/MeasureTheory/OuterMeasure/Caratheodory.lean | 65 | 66 |
/-
Copyright (c) 2024 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.NumberTheory.ModularForms.Basic
import Mathlib.NumberTheory.ModularForms.EisensteinSeries.IsBoundedAtImInfty
import Mathlib.NumberTheory.ModularForms.... | Mathlib/NumberTheory/ModularForms/EisensteinSeries/Basic.lean | 109 | 115 | |
/-
Copyright (c) 2023 Ali Ramsey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ali Ramsey, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.TensorProduct.Finite... | ext : 1
· rw [comp_assoc, fst_comp_inl, comp_id, comp_assoc, comul_comp_inl, ← comp_assoc,
| Mathlib/RingTheory/Coalgebra/Basic.lean | 218 | 219 |
/-
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... | theorem coeff_mul_right' [NonUnitalNonAssocSemiring R] {x y : HahnSeries Γ R} {a : Γ} {s : Set Γ}
(hs : s.IsPWO) (hys : y.support ⊆ s) :
(x * y).coeff a =
∑ ij ∈ addAntidiagonal x.isPWO_support hs a, x.coeff ij.fst * y.coeff ij.snd :=
HahnModule.coeff_smul_right hs hys
@[deprecated (since := "2025-01-3... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 418 | 426 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.PeakFunction
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
/-!
# Fourier inversion formula
In a fin... | theorem MeasureTheory.Integrable.fourier_inversion_inv
(hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V}
(hv : ContinuousAt f v) : 𝓕 (𝓕⁻ f) v = f v := by
rw [fourierIntegralInv_comm]
exact fourier_inversion hf h'f hv
| Mathlib/Analysis/Fourier/Inversion.lean | 177 | 181 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.GameAdd
import Mathlib.SetTheory.Cardinal.Order
import Mathlib.Tactic.AdaptationNo... | rw [single_apply] at hs
split_ifs at hs with hik
swap
· exact (hbot hs).elim
subst hik
classical
refine Lex.acc_of_single hbot x fun j hj ↦ ?_
obtain rfl | hij := eq_or_ne i j
· exact ha _ hs
by_cases h : r j i
· rw [hr j h, single_eq_of_ne hij, single_zero]
exact Lex.acc_zero hbot... | Mathlib/Data/DFinsupp/WellFounded.lean | 134 | 153 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | ContinuousLinearMap.opNorm_le_bound _ hC (norm_setToL1_le_mul_norm hT hC)
theorem norm_setToL1_le' (hT : DominatedFinMeasAdditive μ T C) : ‖setToL1 hT‖ ≤ max C 0 :=
ContinuousLinearMap.opNorm_le_bound _ (le_max_right _ _) (norm_setToL1_le_mul_norm' hT)
theorem setToL1_lipschitz (hT : DominatedFinMeasAdditive μ T ... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 589 | 603 |
/-
Copyright (c) 2022 Mantas Bakšys. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mantas Bakšys
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.Da... | section weak_inequality
variable [PosSMulMono α β] {s : Finset ι} {σ : Perm ι} {f : ι → α} {g : ι → β}
/-- **Rearrangement Inequality**: Pointwise scalar multiplication of `f` and `g` is maximized when
`f` and `g` monovary together on `s`. Stated by permuting the entries of `g`. -/
theorem MonovaryOn.sum_smul_comp_per... | Mathlib/Algebra/Order/Rearrangement.lean | 62 | 108 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 966 | 997 | |
/-
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.CategoryTheory.Comma.Over.Pullback
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import ... | @[reassoc (attr := simp)]
theorem pullbackDiagonalMapIso.inv_snd_snd :
(pullbackDiagonalMapIso f i i₁ i₂).inv ≫ pullback.snd _ _ ≫ pullback.snd _ _ =
pullback.snd _ _ := by
delta pullbackDiagonalMapIso
simp
theorem pullback_fst_map_snd_isPullback :
IsPullback (fst _ _ ≫ i₁ ≫ fst _ _)
| Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 185 | 193 |
/-
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.Pow.Complex
import Qq
/-! # P... | log, norm_neg, arg_ofReal_of_neg hlt, ← ofReal_neg, arg_ofReal_of_nonneg (neg_nonneg.2 hx),
ofReal_zero, zero_mul, add_zero]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 261 | 263 |
/-
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.MeasureTheory.Covering.Differentiation
/-!
# Besicovitch covering theorems
The topological Besicovitch covering theorem ensures that, in a nice... | exact this.mono ball_subset_interior_closedBall
let v : Fin N → Set α := fun i => ⋃ (x : s) (_ : x ∈ u i), closedBall x (r x)
have A : s = ⋃ i : Fin N, s ∩ v i := by
refine Subset.antisymm ?_ (iUnion_subset fun i => inter_subset_left)
intro x hx
obtain ⟨i, y, hxy, h'⟩ :
∃ (i : Fin N) (i_1 : ... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 563 | 686 |
/-
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
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... | have hw (i) (_ : i ∈ t) : w i • p i ≠ 0 → w i ≠ 0 := by aesop
have hw' (i) (_ : i ∈ t) : w i • f (p i) ≠ 0 → w i ≠ 0 := by aesop
rw [← sum_filter_of_ne hw, ← sum_filter_of_ne hw', hf.map_sum_eq_iff]
· simp
· simp +contextual [(h₀ _ _).gt_iff_ne]
· rwa [sum_filter_ne_zero]
· simp +contextual [hmem _ _]
/-... | Mathlib/Analysis/Convex/Jensen.lean | 218 | 233 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | | inl hi => simp [hi]
| inr hi => ?_
convert h ⟨i, hi⟩
have : ∀ j, ⨆ (_ : t j = ⊥), t j = ⊥ := fun j ↦ by simp only [iSup_eq_bot, imp_self]
rw [iSup_split _ (fun j ↦ t j = ⊥), iSup_subtype]
simp only [iSup_comm (ι' := _ ≠ i), this, ne_eq, sup_of_le_right, Subtype.mk.injEq, iSup_bot,
| Mathlib/Order/SupIndep.lean | 400 | 405 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
import Mathlib.Logic.Function.Basic
/-!
# Shapes of homological complexes
We define a st... |
lemma prev_eq_self' (c : ComplexShape ι) (j : ι) (hj : ∀ i, ¬ c.Rel i j) :
c.prev j = j :=
dif_neg (by simpa using hj)
| Mathlib/Algebra/Homology/ComplexShape.lean | 154 | 158 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,201 | 1,205 | |
/-
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.Topology.Separation.Hausdorff
import Mathlib.Topology.Clopen
/-!
# Priestley spaces
This file defines Priestley spaces. A Priestley space is an ordered c... | exact ⟨U, hU, Or.inr hU', hx, hy⟩
-- See note [lower instance priority]
instance (priority := 100) PriestleySpace.toT2Space : T2Space α :=
⟨fun _ _ h ↦
let ⟨U, hU, _, hx, hy⟩ := exists_isClopen_upper_or_lower_of_ne h
| Mathlib/Topology/Order/Priestley.lean | 67 | 72 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | rw [mem_span_singleton'] at hu hv
rw [← hu.choose_spec, ← hv.choose_spec] at h1
exact ⟨_, _, h1⟩
· exact fun ⟨u, v, h1⟩ =>
| Mathlib/RingTheory/Ideal/Operations.lean | 533 | 536 |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... |
/-- The edist to a set is bounded above by the edist to any of its points -/
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 95 | 96 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | theorem sum_modByMonic_coeff (hq : q.Monic) {n : ℕ} (hn : q.degree ≤ n) :
(∑ i : Fin n, monomial i ((p %ₘ q).coeff i)) = p %ₘ q := by
nontriviality R
exact
(sum_fin (fun i c => monomial i c) (by simp) ((degree_modByMonic_lt _ hq).trans_le hn)).trans
(sum_monomial_eq _)
| Mathlib/Algebra/Polynomial/Div.lean | 424 | 430 |
/-
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.Data.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... | theorem powersetCardAux_eq_map_coe {n} {l : List α} :
powersetCardAux n l = (sublistsLen n l).map (↑) := by
rw [powersetCardAux, sublistsLenAux_eq, append_nil]
@[simp]
| Mathlib/Data/Multiset/Powerset.lean | 154 | 158 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... |
end PNat
| Mathlib/Data/PNat/Factors.lean | 215 | 216 |
/-
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 | 456 | 461 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 658 | 660 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
/-!
# The category of functors and natural transf... | def rightUnitor (F : C ⥤ D) :
F ⋙ 𝟭 D ≅ F where
hom := { app := fun X => 𝟙 (F.obj X) }
| Mathlib/CategoryTheory/Functor/Category.lean | 160 | 162 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | theorem mem_memberSubfamily : s ∈ 𝒜.memberSubfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := by
simp_rw [memberSubfamily, mem_image, mem_filter]
| Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 56 | 57 |
/-
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.Pi
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite.Basic
/-!
# Fintype instances for pi types
-/
assert_not_exists Order... |
@[simp]
| Mathlib/Data/Fintype/Pi.lean | 58 | 59 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,174 | 1,175 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | theorem orderOf_dvd_of_mem_zpowers (h : y ∈ Subgroup.zpowers x) : orderOf y ∣ orderOf x := by
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp h
rw [orderOf_dvd_iff_pow_eq_one]
exact zpow_pow_orderOf
theorem smul_eq_self_of_mem_zpowers {α : Type*} [MulAction G α] (hx : x ∈ Subgroup.zpowers y)
{a : α} (hs : y •... | Mathlib/GroupTheory/OrderOfElement.lean | 641 | 648 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Batteries.Tactic.Congr
/-... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 1,016 | 1,020 |
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