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) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | obtain ⟨h, hn1, hnh⟩ := Cardinal.three_le hcard 1 w.head⁻¹
have hnot1 : h * w.head ≠ 1 := by
| Mathlib/GroupTheory/CoprodI.lean | 856 | 857 |
/-
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.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | apply (intervalIntegrable_congr_codiscreteWithin _).1 hf₁
rw [eventuallyEq_iff_exists_mem]
exact ⟨(circleMap c R)⁻¹' {z | f₁ z = f₂ z},
| Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 207 | 209 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma Args.numeric_P1 {x y} : (Args.P1 x y).Numeric ↔ x.Numeric ∧ y.Numeric := by
simp [Args.Numeric, Args.toMultiset]
| Mathlib/SetTheory/Surreal/Multiplication.lean | 165 | 166 |
/-
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.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... | @[deprecated (since := "2025-04-06")]
alias coeLinearMap_eq_dfinsupp_sum := coeLinearMap_eq_dfinsuppSum
| Mathlib/Algebra/DirectSum/Module.lean | 386 | 387 |
/-
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.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | theorem image₂_image_right (f : α → γ → δ) (g : β → γ) :
image₂ f s (t.image g) = image₂ (fun a b => f a (g b)) s t :=
coe_injective <| by
push_cast
| Mathlib/Data/Finset/NAry.lean | 269 | 272 |
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | erw [LinearMap.mem_ker] at hker
constructor
· simp [hx]
· intro y _
rw [IsOrtho, b]
change (B.domRestrict W) ⟨x, hx⟩ y = 0
rw [hker]
rfl
· simp_rw [Submodule.mem_map, LinearMap.mem_ker]
refine ⟨⟨x, hx.1⟩, ?_, rfl⟩
ext y
change B x y = 0
rw [b]
exact hx.2 _ S... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 258 | 275 |
/-
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.Multiset.Dedup
import Mathlib.Data.List.Infix
/-!
# Preparations for defining operations on `Finset`.
The operations here ignore multiplicities,... | (le_ndinter.1 le_rfl).2
| Mathlib/Data/Multiset/FinsetOps.lean | 231 | 232 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | @[simp] lemma ofReal_comp_balance {ι : Type*} [Fintype ι] (f : ι → ℝ) :
ofReal ∘ balance f = balance (ofReal ∘ f : ι → K) := funext <| ofReal_balance _
| Mathlib/Analysis/RCLike/Basic.lean | 253 | 254 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,395 | 2,397 | |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | lemma indicator_eq_zero_iff_not_mem : indicator s 1 i = (0 : M₀) ↔ i ∉ s := by
classical simp [indicator_apply, imp_false]
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 70 | 71 |
/-
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... | measure_subtype_coe_le_comap hu t
theorem volume_subtype_coe_eq_zero_of_volume_eq_zero (hu : NullMeasurableSet u) {t : Set u}
(ht : volume t = 0) : volume (((↑) : u → δ) '' t) = 0 :=
| Mathlib/MeasureTheory/Measure/Restrict.lean | 764 | 767 |
/-
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.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | have h₀ : ‖x‖ ≠ 0 := by
rw [norm_ne_zero_iff]
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 569 | 570 |
/-
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 ... | · simp [Category.assoc, lift_snd, condition_assoc, condition]
· simp only [Category.assoc, lift_fst_assoc, lift_snd, lift_fst, Category.id_comp]
· apply pullback.hom_ext
· apply pullback.hom_ext
· simp only [Category.assoc, lift_snd_assoc, lift_fst_assoc, lift_fst, Category.id_comp]
... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 432 | 439 |
/-
Copyright (c) 2023 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Fangming Li
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Fintype.Pigeonhole
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sigma
import Mathlib.... | exact ⟨p 0⟩
| Mathlib/Order/RelSeries.lean | 177 | 178 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Ma... | end Ext
section
variable (R φ)
| Mathlib/LinearAlgebra/Pi.lean | 260 | 264 |
/-
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... | not_nontrivial_iff_subsingleton, card_le_one_iff_subsingleton_coe, coe_sort_coe]
theorem exists_ne_of_one_lt_card (hs : 1 < #s) (a : α) : ∃ b, b ∈ s ∧ b ≠ a := by
| Mathlib/Data/Finset/Card.lean | 654 | 656 |
/-
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.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- The oriented angle between three points equals the unoriented angle if the sign is
positive. -/
theorem oangle_eq_angle_of_sign_eq_one {p₁ p₂ p₃ : P} (h : (∡ p₁ p₂ p₃).sign = 1) :
∡ p₁ p₂ p₃ = ∠ p₁ p₂ p₃ :=
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 336 | 340 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Uthaiwat, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.I... | @[simp] lemma isNilpotent_reflect_iff {P : R[X]} {N : ℕ} (hN : P.natDegree ≤ N) :
IsNilpotent (reflect N P) ↔ IsNilpotent P := by
simp only [Polynomial.isNilpotent_iff, coeff_reverse]
refine ⟨fun h i ↦ ?_, fun h i ↦ ?_⟩ <;> rcases le_or_lt i N with hi | hi
· simpa [tsub_tsub_cancel_of_le hi] using h (N - i)
... | Mathlib/RingTheory/Polynomial/Nilpotent.lean | 91 | 98 |
/-
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.Polynomial.AlgebraMap
import Mathlib.RingTheory.IsTensorProduct
/-!
# Base change of polynomial algebras
Given `[CommSemiring R] [Semiring A] [Al... | Mathlib/RingTheory/PolynomialAlgebra.lean | 257 | 267 | |
/-
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... | rw [← hl]
apply nodup_nil
clear hl
| Mathlib/Data/List/Sigma.lean | 595 | 597 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 177 | 181 | |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | simp_rw [← pp.pow_dvd_iff_le_factorization hn]
exact h p _ pp (ordProj_dvd _ _)
theorem prod_primeFactors_dvd (n : ℕ) : ∏ p ∈ n.primeFactors, p ∣ n := by
by_cases hn : n = 0
| Mathlib/Data/Nat/Factorization/Basic.lean | 361 | 365 |
/-
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
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 382 | 383 | |
/-
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.Algebra.CharP.Basic
import Mathlib.ModelTheory.Algebra.Ring.FreeCommRing
import Mathlib.ModelTheory.Algebra.Field.Basic
/-!
# First order theory of field... | simp [eqZero, Term.realize]
/-- The first order theory of fields of characteristic `p` as a theory over the language of rings -/
| Mathlib/ModelTheory/Algebra/Field/CharP.lean | 38 | 40 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | ring
· field_simp
· field_simp
theorem ConvexOn.secant_mono_aux2 (hf : ConvexOn 𝕜 s f) {x y z : 𝕜} (hx : x ∈ s) (hz : z ∈ s)
(hxy : x < y) (hyz : y < z) : (f y - f x) / (y - x) ≤ (f z - f x) / (z - x) := by
have hxy' : 0 < y - x := by linarith
have hxz' : 0 < z - x := by linarith
rw [div_le_div_iff... | Mathlib/Analysis/Convex/Slope.lean | 233 | 249 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | (le_iSup (α := ℝ≥0∞) _ N)
· simp only [NNReal.coe_natCast, hf.adapted.integrable_upcrossingsBefore hab]
· exact fun n => measurable_from_top.comp_aemeasurable
(hf.adapted.measurable_upcrossingsBefore hab).aemeasurable
· filter_upwards with ω N M hNM
rw [Nat.cast_le]
exact u... | Mathlib/Probability/Martingale/Upcrossing.lean | 804 | 812 |
/-
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
/... | (ball_smul_closedBall _ _ hr₂)
| Mathlib/Analysis/Seminorm.lean | 845 | 846 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Equiv
/-!
# Partially defined linear operators on Hilbert spaces
We will develop... | (adjoint_apply_of_dense hT x).symm ▸ adjointAux_inner hT x
include hT in
| Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 180 | 182 |
/-
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... | (∀ i, ∃ U, IsOpen U ∧ ∀ x, f x ∈ U ↔ x.1 = i) := by
refine ⟨fun h ↦ ⟨fun i ↦ h.comp IsEmbedding.sigmaMk.1, fun i ↦ ?_⟩, ?_⟩
· rcases h.isOpen_iff.1 (isOpen_range_sigmaMk (i := i)) with ⟨U, hUo, hU⟩
| Mathlib/Topology/Constructions.lean | 1,086 | 1,088 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.GradedObject
/-!
# The action of bifunctors on graded objects
Given a bifunctor `F : C₁ ⥤ C₂ ⥤ C₃` and types `I` and `J`, we construct an obvious... | lemma ι_mapBifunctorMapObjDesc {X : GradedObject I C₁} {Y : GradedObject J C₂} {A : C₃} {k : K}
[HasMap (((mapBifunctor F I J).obj X).obj Y) p]
(f : ∀ (i : I) (j : J) (_ : p ⟨i, j⟩ = k), (F.obj (X i)).obj (Y j) ⟶ A)
(i : I) (j : J) (hij : p ⟨i, j⟩ = k) :
ιMapBifunctorMapObj F p X Y i j k hij ≫ mapBifunc... | Mathlib/CategoryTheory/GradedObject/Bifunctor.lean | 104 | 110 |
/-
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.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.Calculus.LogDeriv... | (h₂ : f x ∈ slitPlane) : HasFDerivAt (fun t => log (f t)) ((f x)⁻¹ • f') x :=
(hasStrictDerivAt_log h₂).hasDerivAt.comp_hasFDerivAt x h₁
| Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | 95 | 97 |
/-
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... | congr! 1
refine iSup_subtype.trans ?_
congr! 1
simp [iSup_and, @iSup_comm _ (_ ∈ s)]
@[deprecated (since := "2024-11-24")]
alias CompleteLattice.independent_iff_supIndep := iSupIndep_iff_supIndep
alias ⟨iSupIndep.supIndep, Finset.SupIndep.independent⟩ := iSupIndep_iff_supIndep
| Mathlib/Order/SupIndep.lean | 489 | 498 |
/-
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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | simp_rw [ht, L.map_zero₂]
simp_rw [convolution_def, h2]
variable [BorelSpace G] [SecondCountableTopology G]
variable [IsAddLeftInvariant μ] [SFinite μ]
/-- Approximate `(f ⋆ g) x₀` if the support of the `f` is bounded within a ball, and `g` is near
`g x₀` on a ball with the same radius around `x₀`. See `dist_... | Mathlib/Analysis/Convolution.lean | 728 | 737 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | theorem whiskerLeft_comp_bimod {X Y Z : Mon_ C} (M : Bimod X Y) {N P Q : Bimod Y Z} (f : N ⟶ P)
(g : P ⟶ Q) : whiskerLeft M (f ≫ g) = whiskerLeft M f ≫ whiskerLeft M g := by
ext
apply Limits.coequalizer.hom_ext
| Mathlib/CategoryTheory/Monoidal/Bimod.lean | 746 | 749 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-! # Gamma distributions over ℝ
Define the gamm... |
/-- The pdf of the gamma distribution integrates to 1 -/
@[simp]
lemma lintegral_gammaPDF_eq_one {a r : ℝ} (ha : 0 < a) (hr : 0 < r) :
∫⁻ x, gammaPDF a r x = 1 := by
have leftSide : ∫⁻ x in Iio 0, gammaPDF a r x = 0 := by
rw [setLIntegral_congr_fun measurableSet_Iio
(ae_of_all _ (fun x (hx : x < 0) ↦ g... | Mathlib/Probability/Distributions/Gamma.lean | 97 | 117 |
/-
Copyright (c) 2021 Alex Kontorovich, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth
-/
import Mathlib.Algebra.Group.Pointwise.Set.Lattice
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Alge... | exact isClosedMap_const
· exact (Homeomorph.smulOfNeZero c hne).isClosedMap
theorem IsClosed.smul₀ {E : Type*} [Zero E] [MulActionWithZero G₀ E] [TopologicalSpace E]
[T1Space E] [ContinuousConstSMul G₀ E] (c : G₀) {s : Set E} (hs : IsClosed s) :
IsClosed (c • s) :=
isClosedMap_smul₀ c s hs
| Mathlib/Topology/Algebra/ConstMulAction.lean | 359 | 366 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | rcases lt_or_gt_of_ne hxa with hxa | hxa
· rcases lt_or_gt_of_ne hya with hya | hya
· convert hf.secant_strict_mono_aux3 hx ha hxy hya using 1 <;> rw [← neg_div_neg_eq] <;>
field_simp
· convert hf.slope_strict_mono_adjacent hx hy hxa hya using 1
rw [← neg_div_neg_eq]; field_simp
| Mathlib/Analysis/Convex/Slope.lean | 302 | 307 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | theorem toBlocks_fromBlocks₁₂ (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) : (fromBlocks A B C D).toBlocks₁₂ = B :=
rfl
| Mathlib/Data/Matrix/Block.lean | 97 | 100 |
/-
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 | 1,339 | 1,377 | |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | /-! ### The univariate polynomials `ΨSqₙ` -/
/-- The univariate polynomials `ΨSqₙ` congruent to `ψₙ²`. -/
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 251 | 253 |
/-
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.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 311 | 312 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | simp only [isAtomic_iff, f.surjective.forall, f.surjective.exists, ← map_bot f, f.eq_iff_eq,
f.le_iff_le, f.isAtom_iff]
| Mathlib/Order/Atoms.lean | 1,006 | 1,007 |
/-
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 | 1,276 | 1,277 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,009 | 2,015 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | haveI : Nonempty X' := ⟨eX x⟩
haveI : Nonempty Y := ⟨y⟩
| Mathlib/Topology/PartialHomeomorph.lean | 928 | 929 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | · exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩
theorem IsConnected.preimage_of_isOpenMap [TopologicalSpace β] {s : Set β} (hs : IsConnected s)
{f : α → β} (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
IsConnected (f ⁻¹' s) :=
⟨hs.nonempty.preimage' hsf, hs.isPreconnected.preimag... | Mathlib/Topology/Connected/Basic.lean | 361 | 370 |
/-
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.Topology.UniformSpace.Basic
import Mathlib.Topology.Compactness.Compact
/-!
# Compact sets in uniform spaces
* `compa... | Mathlib/Topology/UniformSpace/Compact.lean | 273 | 277 | |
/-
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.RiemannZeta
import Mathlib.NumberTheory.Harmonic.GammaDeriv
/-!
# Asymptotics of `ζ s` as `s → 1`
The goal of this file is to ev... | term n 1 = (log (n + 1) - log n) - 1 / (n + 1) := by
have hv : ∀ x ∈ uIcc (n : ℝ) (n + 1), 0 < x := by
intro x hx
rw [uIcc_of_le (by simp only [le_add_iff_nonneg_right, zero_le_one])] at hx
exact (Nat.cast_pos.mpr hn).trans_le hx.1
calc term n 1
_ = ∫ x : ℝ in n..(n + 1), (x - n) / x ^ 2 := by
... | Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean | 73 | 110 |
/-
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... |
instance : FaithfulSMul (PushoutI φ) (NormalWord d) :=
⟨fun h => by simpa using congr_arg prod (h empty)⟩
| Mathlib/GroupTheory/PushoutI.lean | 570 | 573 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.Ring.Opposite
import Ma... |
/-- `Mat_.lift F` is the unique additive functor `L : Mat_ C ⥤ D` such that `F ≅ embedding C ⋙ L`.
-/
def liftUnique (F : C ⥤ D) [Functor.Additive F] (L : Mat_ C ⥤ D) [Functor.Additive L]
(α : embedding C ⋙ L ≅ F) : L ≅ lift F :=
NatIso.ofComponents
| Mathlib/CategoryTheory/Preadditive/Mat.lean | 421 | 426 |
/-
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... | suffices cexp (π * I * n ^ 2 * 2 : ℂ) = 1 by rw [mul_add, Complex.exp_add, this, mul_one]
rw [(by push_cast; ring : (π * I * n ^ 2 * 2 : ℂ) = (n ^ 2 :) * (2 * π * I)), exp_int_mul,
exp_two_pi_mul_I, one_zpow]
lemma jacobiTheta₂'_add_left (z τ : ℂ) : jacobiTheta₂' (z + 1) τ = jacobiTheta₂' z τ := by
| Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 421 | 425 |
/-
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 | 1,269 | 1,273 | |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | Mathlib/Data/List/Indexes.lean | 267 | 277 | |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 621 | 625 | |
/-
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.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | rw [mem_roots (map_ne_zero hp), IsRoot, Polynomial.eval_map]
theorem rootSet_monomial [CommRing S] [IsDomain S] [Algebra R S] {n : ℕ} (hn : n ≠ 0) {a : R}
| Mathlib/Algebra/Polynomial/FieldDivision.lean | 469 | 471 |
/-
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, Alex Keizer
-/
import Mathlib.Data.List.OfFn
import Batteries.Data.List.Perm
import Mathlib.Data.List.Nodup
/-!
# Lists of elements of `Fin n`... | rw [finRange_eq_pmap_range]
exact (Pairwise.pmap nodup_range _) fun _ _ _ _ => @Fin.ne_of_val_ne _ ⟨_, _⟩ ⟨_, _⟩
@[simp]
| Mathlib/Data/List/FinRange.lean | 37 | 40 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | · rw [nodup_iff_injective_get]
rintro ⟨i, hi⟩ ⟨j, hj⟩ h
| Mathlib/Data/List/Rotate.lean | 593 | 594 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson
-/
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
/-!
# Nondeterministic Finite Automata
This file contains the definition of a Nondeterministic Finite... | step s a := {M.step s a}
start := {M.start}
accept := M.accept
@[simp]
theorem toNFA_evalFrom_match (M : DFA α σ) (start : σ) (s : List α) :
M.toNFA.evalFrom {start} s = {M.evalFrom start s} := by
| Mathlib/Computability/NFA.lean | 125 | 131 |
/-
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 | 1,399 | 1,422 | |
/-
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.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
/-!
# Image of a `Finset α` under a partially defined function
In this file we... | theorem pimage_subset {t : Finset β} : s.pimage f ⊆ t ↔ ∀ x ∈ s, ∀ y ∈ f x, y ∈ t := by
simp [subset_iff, @forall_swap _ β]
@[mono]
theorem pimage_mono (h : s ⊆ t) : s.pimage f ⊆ t.pimage f :=
pimage_subset.2 fun x hx _ hy => mem_pimage.2 ⟨x, h hx, hy⟩
theorem pimage_inter [DecidableEq α] : (s ∩ t).pimage f ⊆ s.p... | Mathlib/Data/Finset/PImage.lean | 90 | 97 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | theorem prod_trans (e : PartialHomeomorph X Y) (f : PartialHomeomorph Y Z)
(e' : PartialHomeomorph X' Y') (f' : PartialHomeomorph Y' Z') :
(e.prod e').trans (f.prod f') = (e.trans f).prod (e'.trans f') :=
| Mathlib/Topology/PartialHomeomorph.lean | 919 | 921 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.TensorProduct.Associator
/-!
# The `A`-module structure on `M ⊗[R] N`
Whe... | -- explicit `Eq.refl`s here help with performance, but also make it clear that the `ext` are
-- letting us prove the result as an equality of pure tensors.
(TensorProduct.ext <| ext fun m q => LinearMap.ext fun p => Eq.refl <|
(m ⊗ₜ[R] q) ⊗ₜ[A] p)
(curry_injective <| TensorProduct.ext' fun m p => ... | Mathlib/LinearAlgebra/TensorProduct/Tower.lean | 480 | 487 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | lhs
congr
rfl
rw [Cofix.mk]
rw [Cofix.dest_corec]
rw [← comp_map, ← appendFun_comp, id_comp]
rw [← comp_map, ← appendFun_comp, id_comp, ← Cofix.mk]
congr
apply congrArg
funext x
apply Quot.sound
rfl
theorem Cofix.dest_mk {α : TypeVec n} (x : F (α.append1 <| Cofix F α)) :
Cofix.des... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 314 | 332 |
/-
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.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | map_eq := by rw [Measure.map_add _ _ hf.measurable, hf.map_eq, hf'.map_eq]
theorem smul_measure {R : Type*} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] {f : α → β}
(hf : MeasurePreserving f μa μb) (c : R) : MeasurePreserving f (c • μa) (c • μb) where
measurable := hf.measurable
map_eq := by rw [Measure.map_smu... | Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 163 | 175 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | convert← this
rw [← mul_neg_geom_sum, Finset.mul_sum]
theorem geom_series_succ (x : R) (h : ‖x‖ < 1) : ∑' i : ℕ, x ^ (i + 1) = ∑' i : ℕ, x ^ i - 1 := by
rw [eq_sub_iff_add_eq, (summable_geometric_of_norm_lt_one h).tsum_eq_zero_add,
| Mathlib/Analysis/SpecificLimits/Normed.lean | 270 | 274 |
/-
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.Hom.Basic
/-!
# Unbounded lattice homomorphisms
This file defines unbounded lattice homomorphisms. _Bounded_ lattice homomorphisms are defined in
`... | instance : SemilatticeSup (SupHom α β) :=
(DFunLike.coe_injective.semilatticeSup _) fun _ _ => rfl
| Mathlib/Order/Hom/Lattice.lean | 291 | 292 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Lie.Weights.Killing
import Mathlib.LinearAlgebra.RootSystem.Basic
import Mathlib.LinearAlgebra.RootSystem.Reduced
import Mathlib.LinearAlgebra.RootSy... | lemma chainLength_zero [Nontrivial L] : chainLength 0 β = 0 := by
simp [← chainBotCoeff_add_chainTopCoeff]
/-- If `β - qα ... β ... β + rα` is the `α`-chain through `β`, then
`β (coroot α) = q - r`. In particular, it is an integer. -/
| Mathlib/Algebra/Lie/Weights/RootSystem.lean | 181 | 185 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Group.Submonoid.DistribMulAction
import Mathlib.GroupTheory.OreLocalization.Basic
import Mathlib.Algebra.GroupWi... | use sd, 1
simp only [one_smul, one_mul, mul_smul, ← hb, Submonoid.smul_def, ← mul_assoc, and_true]
| Mathlib/RingTheory/OreLocalization/Basic.lean | 132 | 133 |
/-
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.Gaussian.FourierTransform
import Mathlib.Analysis.Fourier.PoissonSummation
/-!
# Poisson summation applied to the Gaussian
... | rw [cocompact_eq_atBot_atTop, isLittleO_sup]
constructor
· refine ((cexp_neg_quadratic_isLittleO_rpow_atTop ha (-b) s).comp_tendsto
Filter.tendsto_neg_atBot_atTop).congr' (Eventually.of_forall fun x ↦ ?_) ?_
· simp only [neg_mul, Function.comp_apply, ofReal_neg, neg_sq, mul_neg, neg_neg]
· refine (e... | Mathlib/Analysis/SpecialFunctions/Gaussian/PoissonSummation.lean | 55 | 66 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | simp
· filter_upwards [Ioo_mem_nhdsGT (neg_lt_self pi_div_two_pos)] with x hx
exact cos_pos_of_mem_Ioo hx
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 972 | 974 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | have J : ∀ n : ℕ, dist (g ⟨x, I⟩) y ≤ (1 / 2) ^ n + (1 / 2) ^ n := fun n =>
calc
dist (g ⟨x, I⟩) y ≤ dist (g ⟨x, I⟩) (u (x n)) + dist y (u (x n)) :=
dist_triangle_right _ _ _
_ ≤ (1 / 2) ^ n + (1 / 2) ^ n := add_le_add (A ⟨x, I⟩ n) (hx n)
have L : Tendsto (fun n : ℕ => (1 / 2 : ℝ... | Mathlib/Topology/MetricSpace/PiNat.lean | 722 | 806 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Induction
import Mathlib.Data.List.TakeWhile
/-!
# Dropping or taking from lists on the right
Taking or removing element from the tail e... | · exact drop_length
| Mathlib/Data/List/DropRight.lean | 78 | 78 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... |
@[to_additive existing, deprecated (since := "2025-04-19")]
alias prod_univ_get' := prod_univ_fun_getElem
| Mathlib/Algebra/BigOperators/Fin.lean | 101 | 103 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... | Mathlib/Topology/FiberBundle/Basic.lean | 885 | 888 | |
/-
Copyright (c) 2020 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.Geometry.Manifold.MFDeriv.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | congr
· have : (fun z : M × M' => f (z.1, p.2)) = (fun z : M => f (z, p.2)) ∘ Prod.fst := rfl
rw [this, mfderiv_comp (I' := I)]
· simp only [mfderiv_fst, id_eq]
| Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 543 | 546 |
/-
Copyright (c) 2024 Ira Fesefeldt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ira Fesefeldt
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
/-!
# Ordinal Approximants for the Fixed points on complete lattices
This file sets up the ordinal-indexed approximation t... | · rw [IH a' ha'b (le_of_not_lt haa), h]
· exact lfpApprox_monotone f x h_ab
/-- There are distinct indices smaller than the successor of the domain's cardinality
yielding the same value -/
theorem exists_lfpApprox_eq_lfpApprox : ∃ a < ord <| succ #α, ∃ b < ord <| succ #α,
a ≠ b ∧ lfpApprox f x a = lfpApprox ... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 164 | 173 |
/-
Copyright (c) 2020 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.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | (hgs : ContinuousWithinAt g s x) :
P (f ∘ g ∘ e.symm) (e.symm ⁻¹' s) (e x) ↔ P (f' ∘ g ∘ e'.symm) (e'.symm ⁻¹' s) (e' x) := by
rw [← Function.comp_assoc, hG.liftPropWithinAt_indep_chart_source_aux (f ∘ g) he xe he' xe',
Function.comp_assoc, hG.liftPropWithinAt_indep_chart_target_aux xe' hf xf hf' xf' hgs]... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 273 | 280 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | -/
theorem natDegree_multiset_prod' (h : (t.map fun f => leadingCoeff f).prod ≠ 0) :
t.prod.natDegree = (t.map fun f => natDegree f).sum := by
| Mathlib/Algebra/Polynomial/BigOperators.lean | 165 | 167 |
/-
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.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
/-!
# `min` and `max` in... | @[to_additive max_sub_sub_left]
theorem max_div_div_left' (a b c : α) : max (a / b) (a / c) = a / min b c := by
| Mathlib/Algebra/Order/Group/MinMax.lean | 57 | 58 |
/-
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
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | eq_zero_iff_equiv_zero _ |>.not
theorem norm_const (q : ℚ) : norm (const (padicNorm p) q) = padicNorm p q := by
obtain rfl | hq := eq_or_ne q 0
· simp [norm]
· simp [norm, not_equiv_zero_const_of_nonzero hq]
theorem norm_values_discrete (a : PadicSeq p) (ha : ¬a ≈ 0) : ∃ z : ℤ, a.norm = (p : ℚ) ^ (-z) := by
... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 281 | 295 |
/-
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.Geometry.Euclidean.Angle.Oriented.RightAngle
import Mathlib.Geometry.Euclidean.Circumcenter
/-!
# Angles in circles and sphere.
This file proves results ... | namespace Sphere
/-- The angle at the apex of an isosceles triangle is `π` minus twice a base angle, oriented
angle-at-point form where the apex is given as the center of a circle. -/
theorem oangle_eq_pi_sub_two_zsmul_oangle_center_left {s : Sphere P} {p₁ p₂ : P} (hp₁ : p₁ ∈ s)
(hp₂ : p₂ ∈ s) (h : p₁ ≠ p₂) : ∡ p₁... | Mathlib/Geometry/Euclidean/Angle/Sphere.lean | 108 | 113 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 685 | 697 | |
/-
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... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,188 | 1,193 | |
/-
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 functional equation for the Jacobi theta function: `jacobiTheta₂ z τ` is an explicit factor
times `jacobiTheta₂ (z / τ) (-1 / τ)`. This is the key lemma behind the proof of the functional
equation for L-series of even Dirichlet characters. -/
theorem jacobiTheta₂_functional_equation (z τ : ℂ) : jacobiTheta₂ z ... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 468 | 494 |
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 806 | 806 | |
/-
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.Measure.Trim
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
/-!
# Almost everywhere measurable functions
A funct... | · rintro _ ⟨x, rfl⟩
by_cases hx : x ∈ s
· simpa [g, hx] using h₀.some_mem
· simp only [g, hx, piecewise_eq_of_not_mem, not_false_iff]
contrapose! hx
| Mathlib/MeasureTheory/Measure/AEMeasurable.lean | 187 | 191 |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Ma... | apply Functor.hext
· exact fun i ↦ rfl
· rintro ⟨i⟩ ⟨j⟩ ⟨⟨rfl : i = j⟩⟩
simp [F']
have t₁' := @t₁ (Discrete.functor (fun j ↦ F.obj ⟨j.succ⟩))
(Cofan.mk (pullback c₂.inr i) fun j ↦ pullback.lift (α.app _ ≫ c₁.inj _) (c.ι.app _) ?_)
(Discrete.natTrans fun i ↦ α.app _) (pullback.fst _ _) ?_
... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 589 | 596 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | /-- Cardinality of a Young diagram -/
protected abbrev card (μ : YoungDiagram) : ℕ :=
μ.cells.card
section Transpose
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 174 | 179 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... |
theorem mem_perpBisector_iff_dist_eq' : c ∈ perpBisector p₁ p₂ ↔ dist p₁ c = dist p₂ c := by
simp only [mem_perpBisector_iff_dist_eq, dist_comm]
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 92 | 95 |
/-
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.Nodup
import Mathlib.Data.List.Lattice
import Batteries.Data.List.Pairwise
/-!
# Erasure of duplicates in a list
This file proves basic res... | @[simp]
theorem dedup_eq_nil (l : List α) : l.dedup = [] ↔ l = [] := by
induction l with
| nil => exact Iff.rfl
| cons a l hl =>
by_cases h : a ∈ l
· simp only [List.dedup_cons_of_mem h, hl, List.ne_nil_of_mem h, reduceCtorEq]
· simp only [List.dedup_cons_of_not_mem h, List.cons_ne_nil]
protected the... | Mathlib/Data/List/Dedup.lean | 94 | 105 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | /-- **Cauchy-Schwarz inequality** for finsets using square roots in `ℝ≥0`. -/
lemma sum_mul_le_sqrt_mul_sqrt (s : Finset ι) (f g : ι → ℝ≥0) :
| Mathlib/Data/Real/Sqrt.lean | 416 | 417 |
/-
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.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | lemma arg_lt_pi_div_two_iff {z : ℂ} : arg z < π / 2 ↔ 0 < re z ∨ im z < 0 ∨ z = 0 := by
rw [lt_iff_le_and_ne, arg_le_pi_div_two_iff, Ne, arg_eq_pi_div_two_iff]
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 371 | 372 |
/-
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.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 872 | 874 | |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 183 | 194 | |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | (IsAlgClosed.splits_codomain _) (minpoly.natDegree_le x) (fun z hz => ?_) i
classical
rw [← Multiset.mem_toFinset] at hz
obtain ⟨φ, rfl⟩ := (range_eval_eq_rootSet_minpoly K A x).symm.subset hz
exact h φ
variable (K A)
/-- Let `B` be a real number. The set of algebraic integers in `K` whose conjugates ar... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 89 | 98 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Action.Center
import Mathlib.GroupTheory.ClassEquation
import Mathlib.RingTheory.Polynomial.Cyclotomic.Eval
/-!
... | private theorem center_eq_top [Finite D] (hD : InductionHyp D) : Subring.center D = ⊤ := by
classical
cases nonempty_fintype D
set Z := Subring.center D
-- We proceed by contradiction; that is, we assume the center is strictly smaller than `D`.
by_contra! hZ
letI : Field Z := hD.field hZ.lt_top
set q := c... | Mathlib/RingTheory/LittleWedderburn.lean | 61 | 134 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... | theorem isCauSeq_iff_lift {f : ℕ → ℚ} : IsCauSeq abs f ↔ IsCauSeq abs fun i => (f i : ℝ) where
mp H ε ε0 :=
let ⟨δ, δ0, δε⟩ := exists_pos_rat_lt ε0
(H _ δ0).imp fun i hi j ij => by dsimp; exact lt_trans (mod_cast hi _ ij) δε
mpr H ε ε0 :=
(H _ (Rat.cast_pos.2 ε0)).imp fun i hi j ij => by dsimp at hi; ex... | Mathlib/Data/Real/Archimedean.lean | 33 | 38 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... | @[deprecated (since := "2025-01-26")]
alias slope_lt_left_deriv := slope_lt_leftDeriv
/-- If `f : ℝ → ℝ` is strictly convex on `S` and differentiable within `S` at `y ∈ S`, then the
slope of any secant line with right endpoint at `y` is strictly less than the derivative of `f`
| Mathlib/Analysis/Convex/Deriv.lean | 807 | 811 |
/-
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... | letI := Classical.decEq R
if h : Monic (0 : R[X]) then by
haveI := monic_zero_iff_subsingleton.mp h
simp [eq_iff_true_of_subsingleton]
else by unfold divByMonic divModByMonicAux; rw [dif_neg h]
| Mathlib/Algebra/Polynomial/Div.lean | 196 | 201 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... |
/-- The unit sphere in an `n + 1`-dimensional inner product space `E` is an analytic manifold,
| Mathlib/Geometry/Manifold/Instances/Sphere.lean | 368 | 369 |
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