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) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... | Mathlib/Data/DFinsupp/NeLocus.lean | 175 | 176 | |
/-
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.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... | ContinuousWithinAt f s x' ∧
DifferentiableWithinAt 𝕜 (extChartAt I' y ∘ f ∘ (extChartAt I x).symm)
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 328 | 329 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... |
section CommSemiring
variable [CommSemiring R] [CommSemiring A]
variable [Algebra R A] {s t : Set A}
variable (R s t)
theorem adjoin_union_eq_adjoin_adjoin :
adjoin R (s ∪ t) = (adjoin (adjoin R s) t).restrictScalars R := by
simpa using adjoin_algebraMap_image_union_eq_adjoin_adjoin R s t
variable {R}
| Mathlib/RingTheory/Adjoin/Basic.lean | 118 | 130 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... | have hcG := to_subgroup (of_card hGpn) (center G)
rcases iff_card.1 hcG with _
haveI : Nontrivial G := (nontrivial_iff_card <| of_card hGpn).2 ⟨n, hn, hGpn⟩
exact (nontrivial_iff_card hcG).mp (center_nontrivial (of_card hGpn))
/-- The quotient by the center of a group of cardinality `p ^ 2` is cyclic. -/
theor... | Mathlib/GroupTheory/PGroup.lean | 348 | 358 |
/-
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.Continuity
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Pointwise
/-!
#... | hC _ <| Set.mem_image_of_mem _ hx
@[to_additive]
lemma HasCompactMulSupport.exists_bound_of_continuous [TopologicalSpace α]
| Mathlib/Analysis/Normed/Group/Bounded.lean | 96 | 99 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | · intro h
rw [← Category.comp_id f, h, comp_zero]
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 197 | 198 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,553 | 2,555 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... | (lsum ℕ fun i => (p i).subtype : _ →ₗ[R] _).comp
((mapRange.linearMap fun i => LinearMap.toSpanSingleton R (↥(p i)) ⟨v i, hv i⟩ :
_ →ₗ[R] _).comp
(finsuppLequivDFinsupp R : (ι →₀ R) ≃ₗ[R] _).toLinearMap) =
Finsupp.linearCombination R v := by
ext
simp
end Semiring
section ... | Mathlib/LinearAlgebra/DFinsupp.lean | 495 | 511 |
/-
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.Complex
/-!
# The `arctan` function.
Inequalit... | theorem arctan_strictMono : StrictMono arctan := tanOrderIso.symm.strictMono
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 146 | 147 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.NormNum
/-!
# Convex and concave functions
This... |
variable [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommMonoid E]
| Mathlib/Analysis/Convex/Function.lean | 956 | 958 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,621 | 2,621 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,411 | 1,413 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... |
theorem trim_eq_trim_iff {m₁ m₂ : OuterMeasure α} :
m₁.trim = m₂.trim ↔ ∀ s, MeasurableSet s → m₁ s = m₂ s := by
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 336 | 338 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | filter_upwards with v
exact (norm_fourierPowSMulRight_le L f v n).trans (le_of_eq (by ring))
lemma hasFTaylorSeriesUpTo_fourierIntegral {N : WithTop ℕ∞}
(hf : ∀ (n : ℕ), n ≤ N → Integrable (fun v ↦ ‖v‖^n * ‖f v‖) μ)
(h'f : AEStronglyMeasurable f μ) :
HasFTaylorSeriesUpTo N (fourierIntegral 𝐞 μ L.toLin... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 440 | 486 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | bit_negSucc]
| Mathlib/Data/Int/Bitwise.lean | 289 | 289 |
/-
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... |
theorem det_of_card_zero (h : Fintype.card n = 0) (M : Matrix n n R) : M.det = 1 := by
rw [Fintype.card_eq_zero_iff] at h
suffices M = 1 by simp [this]
ext i
exact h.elim i
| Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 81 | 86 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | /-!
### Sets acting on Submodules
Let `R` be a (semi)ring and `M` an `R`-module. Let `S` be a monoid which acts on `M` distributively,
| Mathlib/Algebra/Module/Submodule/Pointwise.lean | 271 | 274 |
/-
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... | /-- A set is a neighborhood of `a` if and only if it contains an interval `(l, u)` containing `a`.
-/
theorem mem_nhds_iff_exists_Ioo_subset [OrderTopology α] [NoMaxOrder α] [NoMinOrder α] {a : α}
{s : Set α} : s ∈ 𝓝 a ↔ ∃ l u, a ∈ Ioo l u ∧ Ioo l u ⊆ s :=
mem_nhds_iff_exists_Ioo_subset' (exists_lt a) (exists_gt... | Mathlib/Topology/Order/Basic.lean | 464 | 469 |
/-
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.Data.Finite.Sum
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutations on... | exact ⟨b, rfl⟩
let σ₁' := subtypePermOfFintype σ h1
let σ₂' := subtypePermOfFintype σ h3
let σ₁ := permCongr (Equiv.ofInjective _ Sum.inl_injective).symm σ₁'
let σ₂ := permCongr (Equiv.ofInjective _ Sum.inr_injective).symm σ₂'
rw [MonoidHom.mem_range, Prod.exists]
use σ₁, σ₂
rw [Perm.s... | Mathlib/GroupTheory/Perm/Finite.lean | 132 | 163 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... |
It is possible to make the hypotheses in `h_add` a bit stronger, and such conditions can be added
once we need them (for example it is only necessary to consider the case where `g` is a multiple
of a characteristic function, and that this multiple doesn't appear in the image of `f`).
To use in an induction proof, the... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,143 | 1,148 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | Mathlib/Data/Set/NAry.lean | 367 | 371 | |
/-
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... | theorem mem_setOf {a : α} {p : α → Prop} : a ∈ { x | p x } ↔ p a :=
Iff.rfl
| Mathlib/Data/Set/Basic.lean | 204 | 205 |
/-
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... | · exact (le_rpow_iff_log_le hx hy).2 h
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 781 | 782 |
/-
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... | have A : Continuous fun y => edist x y := continuous_const.edist continuous_id
obtain ⟨y, ys, hy⟩ := hs.exists_isMinOn hne A.continuousOn
exact ⟨y, ys, le_antisymm (infEdist_le_edist_of_mem ys) (by rwa [le_infEdist])⟩
theorem exists_pos_forall_lt_edist (hs : IsCompact s) (ht : IsClosed t) (hst : Disjoint s t) :
... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 226 | 248 |
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepa... | [CountablyGenerated α] : generateFrom (range (natGeneratingSequence _)) = m := by
rw [natGeneratingSequence, range_enumerateCountable_of_mem _ empty_mem_countableGeneratingSet,
generateFrom_countableGeneratingSet]
| Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 86 | 89 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | simp only [← isOpen_compl_iff, compl_sUnion] at hS ⊢; exact isOpen_sInter <| forall_mem_image.2 hS
| Mathlib/Topology/AlexandrovDiscrete.lean | 71 | 72 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Pointwise.Set.Card
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Measure.Prod
import Mathlib.Topology.Algebr... | @[to_additive (attr := simp)]
theorem inv_apply (μ : Measure G) (s : Set G) : μ.inv s = μ s⁻¹ :=
(MeasurableEquiv.inv G).map_apply s
@[to_additive (attr := simp)]
protected theorem inv_inv (μ : Measure G) : μ.inv.inv = μ :=
| Mathlib/MeasureTheory/Group/Measure.lean | 326 | 331 |
/-
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.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,060 | 1,069 | |
/-
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.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | ⟨some (ret q), v, Tape.mk' ∅ (addBottom L)⟩ := by
induction' n with n IH; · rfl
refine Reaches₀.head ?_ IH
simp only [Option.mem_def, TM1.step]
rw [Option.some_inj, tr, TM1.stepAux, Tape.move_right_n_head, Tape.mk'_nth_nat,
addBottom_nth_succ_fst, TM1.stepAux, iterate_succ', Function.comp_apply, Tape.mo... | Mathlib/Computability/TuringMachine.lean | 644 | 652 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Logic.Small.Basic
import Mathlib.RingTheory.Ideal.Defs
/-!
# Injecti... | def ExtensionOfMaxAdjoin.extendIdealTo (h : Module.Baer R Q) (y : N) : R →ₗ[R] Q :=
(h (ExtensionOfMaxAdjoin.ideal i f y) (ExtensionOfMaxAdjoin.idealTo i f y)).choose
theorem ExtensionOfMaxAdjoin.extendIdealTo_is_extension (h : Module.Baer R Q) (y : N) :
∀ (x : R) (mem : x ∈ ExtensionOfMaxAdjoin.ideal i f y),
... | Mathlib/Algebra/Module/Injective.lean | 249 | 256 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.Measure... | rw [volume_ball, hk, pow_succ (√π), pow_mul, sq_sqrt pi_nonneg, mul_div_assoc, mul_div_assoc]
congr 3
simp? [add_div, add_right_comm, -one_div, Gamma_nat_add_one_add_half] says
| Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 391 | 393 |
/-
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.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | (hg : AEStronglyMeasurable[m] g μ) :
⟪(condExpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, g⟫₂ := by
symm
rw [← sub_eq_zero, ← inner_sub_left, condExpL2]
simp only [mem_lpMeas_iff_aestronglyMeasurable.mpr hg,
Submodule.orthogonalProjection_inner_eq_zero f g]
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 152 | 158 |
/-
Copyright (c) 2018 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
/-!
# Morphisms from equations between objects.
When working categorically, sometimes one encounters an equation `h ... | lemma eqToHom_heq_id_cod (X Y : C) (h : X = Y) : HEq (eqToHom h) (𝟙 Y) := by
subst h; rfl
/-- Two morphisms are conjugate via eqToHom if and only if they are heterogeneously equal.
| Mathlib/CategoryTheory/EqToHom.lean | 59 | 62 |
/-
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... | let ⟨k, hrk⟩ := this _ hym
let ⟨f, hfm, g, hg, hfgrk⟩ := Submodule.mem_sup.1 hrk
| Mathlib/RingTheory/Ideal/Operations.lean | 846 | 847 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 936 | 936 | |
/-
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
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... | @[simp]
theorem map_exp_nhds (x : ℝ) : map exp (𝓝 x) = 𝓝 (exp x) :=
isOpenEmbedding_exp.map_nhds_eq x
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 368 | 371 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.Convex.Segment
import... | theorem Wbtw.trans_right_left {w x y z : P} (h₁ : Wbtw R w x z) (h₂ : Wbtw R x y z) :
Wbtw R w x y := by
rw [wbtw_comm] at *
exact h₁.trans_left_right h₂
| Mathlib/Analysis/Convex/Between.lean | 762 | 766 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | rw [Sphere.mem_coe, mem_sphere, ← mul_self_inj_of_nonneg dist_nonneg (Real.sqrt_nonneg _),
Real.mul_self_sqrt (add_nonneg (mul_self_nonneg _) (mul_self_nonneg _))]
rcases hp₁ with hp₁ | hp₁
· rw [hp₁]
rw [hpo,
dist_sq_smul_orthogonal_vadd_smul_orthogonal_vadd (orthogonalProje... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 76 | 81 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | simp only [and_assoc, and_left_comm, mem_inter, mem_product]
theorem disjoint_product : Disjoint (s ×ˢ t) (s' ×ˢ t') ↔ Disjoint s s' ∨ Disjoint t t' := by
| Mathlib/Data/Finset/Prod.lean | 244 | 246 |
/-
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.PiNat
import Mathlib.Topology.Metrizable.CompletelyMetrizable
import Mathlib.Topology.Sets.Opens
/-!
# Polish spaces
A top... | Mathlib/Topology/MetricSpace/Polish.lean | 375 | 393 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | · rcases insertWith.valid_aux f x hf h.left bl (lt_of_le_not_le h_1 h_2) with ⟨vl, e⟩
suffices H : _ by
refine ⟨vl.balanceL h.right H, ?_⟩
| Mathlib/Data/Ordmap/Ordset.lean | 498 | 500 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | @[simp]
theorem withDensity_eq_zero_iff {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) :
μ.withDensity f = 0 ↔ f =ᵐ[μ] 0 := by
rw [← measure_univ_eq_zero, withDensity_apply _ .univ, restrict_univ, lintegral_eq_zero_iff' hf]
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 216 | 220 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Probability.Kernel.CondDistrib
import Mathlib.Probability.ConditionalProbability
/-!
# Kernel as... | nontriviality Ω
rw [condExpKernel_eq]
have h := aestronglyMeasurable_integral_condDistrib
(aemeasurable_id'' μ (inf_le_right : m ⊓ mΩ ≤ mΩ)) aemeasurable_id
(hf.comp_snd_map_prod_id (inf_le_right : m ⊓ mΩ ≤ mΩ))
rw [MeasurableSpace.comap_id] at h
exact h.mono inf_le_left
| Mathlib/Probability/Kernel/Condexp.lean | 167 | 173 |
/-
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.Order.LeftRight
import Mathlib.Topology.Order.Monotone
/-!
# Left and right limits
We define the (strict) left and right limits of a f... | theorem rightLim_eq_of_tendsto [TopologicalSpace α] [OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[>] a ≠ ⊥) (h' : Tendsto f (𝓝[>] a) (𝓝 y)) :
Function.rightLim f a = y :=
@leftLim_eq_of_tendsto αᵒᵈ _ _ _ _ _ _ f a y h h'
| Mathlib/Topology/Order/LeftRightLim.lean | 75 | 78 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,755 | 1,763 | |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Analysis.Normed.Group.Int
import Mathlib.Analysis.Normed.Group.Subgroup
import Mathlib.Analysis.Normed.Group.Uniform
/-!
# Normed groups homomorphisms... | variable {R R' : Type*} [MonoidWithZero R] [DistribMulAction R V₂] [PseudoMetricSpace R]
[IsBoundedSMul R V₂] [MonoidWithZero R'] [DistribMulAction R' V₂] [PseudoMetricSpace R']
| Mathlib/Analysis/Normed/Group/Hom.lean | 423 | 424 |
/-
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... | @[to_additive]
| Mathlib/Algebra/Order/Group/Abs.lean | 87 | 87 |
/-
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, Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
/-!
# Tensor products of direct sums
This file shows that... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 157 | 161 | |
/-
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... | Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
(atTop_basis.tendsto_iff nhds_basis_ball).trans <| by
simp only [true_and, mem_ball, mem_Ici]
/-- A variant of `tendsto_atTop` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 842 | 847 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Anne Baanen
-/
import Mathlib.Tactic.Ring.Basic
import Mathlib.Tactic.TryThis
import Mathlib.Tactic.Conv
import Mathlib.Util.Qq
/-!
# `ring_nf` tactic
A tactic which ... | (Rat.rawCast (.negOfNat n) d : R) = Int.rawCast (.negOfNat n) / Nat.rawCast d := by simp
| Mathlib/Tactic/Ring/RingNF.lean | 126 | 127 |
/-
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.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... |
/-- Note this matches the convention of `List.ofFn_succ'`, putting the `Fin m` elements first. -/
theorem ofFn_add {m n} (f : Fin (m + n) → α) :
| Mathlib/Data/List/OfFn.lean | 56 | 58 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | Quotient (IsRotated.setoid α)
namespace Cycle
variable {α : Type*}
/-- The coercion from `List α` to `Cycle α` -/
@[coe] def ofList : List α → Cycle α :=
Quot.mk _
instance : Coe (List α) (Cycle α) :=
⟨ofList⟩
@[simp]
theorem coe_eq_coe {l₁ l₂ : List α} : (l₁ : Cycle α) = (l₂ : Cycle α) ↔ l₁ ~r l₂ :=
| Mathlib/Data/List/Cycle.lean | 400 | 414 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | ⟨h.1.mono_ac hle, hle h.2, h.aedisjoint.mono fun _ _ h => hle h⟩
@[to_additive]
theorem preimage_of_equiv {ν : Measure β} (h : IsFundamentalDomain G s μ) {f : β → α}
(hf : QuasiMeasurePreserving f ν μ) {e : G → H} (he : Bijective e)
(hef : ∀ g, Semiconj f (e g • ·) (g • ·)) : IsFundamentalDomain H (f ⁻¹' s) ... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 149 | 156 |
/-
Copyright (c) 2022 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.Comma.Arrow
/-!
# Commutative squares
This file provide an API for commutative squares in categories.
If `top`, `left`, `right`... |
/-- A choice of a diagonal morphism that is part of a `LiftStruct` when
the square has a lift. -/
| Mathlib/CategoryTheory/CommSq.lean | 246 | 248 |
/-
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... | theorem rpow_lt_rpow_of_exponent_neg {x y z : ℝ} (hy : 0 < y) (hxy : y < x) (hz : z < 0) :
x ^ z < y ^ z := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 596 | 597 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,526 | 1,531 | |
/-
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... | Mathlib/Data/List/Rotate.lean | 674 | 681 | |
/-
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.... |
theorem length_ne_zero_of_nontrivial (h : {x | x ∈ s}.Nontrivial) : s.length ≠ 0 :=
fun hs ↦ h.not_subsingleton <| subsingleton_of_length_eq_zero hs
theorem length_pos_of_nontrivial (h : {x | x ∈ s}.Nontrivial) : 0 < s.length :=
Nat.pos_iff_ne_zero.mpr <| length_ne_zero_of_nontrivial h
| Mathlib/Order/RelSeries.lean | 191 | 197 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... | theorem sum_Ico_reflect {δ : Type*} [AddCommMonoid δ] (f : ℕ → δ) (k : ℕ) {m n : ℕ}
(h : m ≤ n + 1) : (∑ j ∈ Ico k m, f (n - j)) = ∑ j ∈ Ico (n + 1 - m) (n + 1 - k), f j :=
@prod_Ico_reflect (Multiplicative δ) _ f k m n h
theorem prod_range_reflect (f : ℕ → M) (n : ℕ) :
(∏ j ∈ range n, f (n - 1 - j)) = ∏ j ∈... | Mathlib/Algebra/BigOperators/Intervals.lean | 220 | 235 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Pi
import Math... | theorem coe_genWeightSpace_of_top (χ : L → R) :
(genWeightSpace M (χ ∘ (⊤ : LieSubalgebra R L).incl) : Submodule R M) = genWeightSpace M χ := by
ext m
simp only [mem_genWeightSpace, LieSubmodule.mem_toSubmodule, Subtype.forall]
apply forall_congr'
simp
@[simp]
| Mathlib/Algebra/Lie/Weights/Basic.lean | 306 | 313 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | · apply Fin.succ_ne_zero
· exact (Fin.succ_injective _).ne hlt.ne
· rw [Fin.lt_iff_val_lt_val]
simpa [this] using hlt
· rw [heq, Fin.cycleRange_self, Fin.succAbove_of_castSucc_lt, swap_apply_right, Fin.castSucc_zero]
| Mathlib/GroupTheory/Perm/Fin.lean | 231 | 235 |
/-
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... | `MetricSpace.induced` useful in case if the domain already has a `UniformSpace` structure. -/
abbrev IsUniformEmbedding.comapMetricSpace {α β} [UniformSpace α] [m : MetricSpace β] (f : α → β)
| Mathlib/Topology/MetricSpace/Basic.lean | 96 | 97 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... |
variable [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
| Mathlib/Analysis/Convex/Basic.lean | 508 | 510 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | Mathlib/Data/TypeVec.lean | 770 | 777 | |
/-
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... | refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.smul c)) _ _ _ _ ?_
ext1 f
suffices c • setToL1 hT f = setToL1SCLM α E μ (hT.smul c) f by rw [← this]; simp [coeToLp]
rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_left c hT]
theorem setToL1_smul_left' (hT : DominatedFinMeasAdditive μ T C)
(hT'... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 475 | 485 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | simp [TerminatedAt]
| Mathlib/Data/Seq/Seq.lean | 704 | 705 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | linarith [show 3 ≤ b ^ 2 - 1 from le_tsub_of_add_le_left (show 4 ≤ b ^ 2 by nlinarith)]
rwa [Nat.mul_div_cancel _ h₂] at h₁
rw [Nat.mul_sub_left_distrib, mul_one, pow_mul]
conv_rhs => rw [← Nat.sub_add_cancel (show 1 ≤ p by omega)]
rw [Nat.pow_succ (b ^ 2)]
suffices h : p * b ^ 2 < (b ^ 2) ^ (p - 1) *... | Mathlib/NumberTheory/FermatPsp.lean | 298 | 329 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | CommRingCat.ofHom (HomogeneousLocalization.mapId 𝒜 (Submonoid.powers_le.mpr x.2)) ≫
(Proj.stalkIso' 𝒜 x.1).toCommRingCatIso.inv ≫
((Proj.toLocallyRingedSpace 𝒜).restrictStalkIso (Opens.isOpenEmbedding _) x).inv := by
rw [awayToΓ, Category.assoc, ← Category.assoc _ (Iso.inv _),
Iso.eq_comp_inv... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 632 | 643 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... |
lemma dual_ne_zero_iff :
dual A K I ≠ 0 ↔ I ≠ 0 := dual_eq_zero_iff.not
| Mathlib/RingTheory/DedekindDomain/Different.lean | 291 | 293 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Basic
/-! # Re... | Mathlib/Analysis/Complex/RealDeriv.lean | 111 | 115 | |
/-
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 | 668 | 670 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Nat
import Ma... |
theorem sq_of_isCoprime {a b c : ℤ} (h : IsCoprime a b) (heq : a * b = c ^ 2) :
∃ a0 : ℤ, a = a0 ^ 2 ∨ a = -a0 ^ 2 :=
sq_of_gcd_eq_one (isCoprime_iff_gcd_eq_one.mp h) heq
@[deprecated (since := "2025-01-23")] alias sq_of_coprime := sq_of_isCoprime
theorem natAbs_euclideanDomain_gcd (a b : ℤ) :
Int.natAbs (... | Mathlib/RingTheory/Int/Basic.lean | 59 | 69 |
/-
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 | 759 | 760 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | protected theorem mul_add (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : φ₁ * (φ₂ + φ₃) = φ₁ * φ₂ + φ₁ * φ₃ :=
ext fun n => by
classical simp only [coeff_mul, mul_add, Finset.sum_add_distrib, LinearMap.map_add]
protected theorem add_mul (φ₁ φ₂ φ₃ : MvPowerSeries σ R) : (φ₁ + φ₂) * φ₃ = φ₁ * φ₃ + φ₂ * φ₃ :=
ext fun n => by
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 277 | 282 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Kim Morrison
-/
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism... | apply id_nonzero X
simp only [← IsIso.hom_inv_id f, h, zero_comp],
fun w => isIso_of_hom_simple w⟩
open scoped Classical in
/-- In any preadditive category with kernels,
the endomorphisms of a simple object form a division ring. -/
| Mathlib/CategoryTheory/Preadditive/Schur.lean | 52 | 58 |
/-
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... | theorem mapIdxMAux'_eq_mapIdxMGo {α} (f : ℕ → α → m PUnit) (as : List α) (arr : Array PUnit) :
mapIdxMAux' f arr.size as = mapIdxM.go f as arr *> pure PUnit.unit := by
revert arr
induction' as with head tail ih <;> intro arr
· simp only [mapIdxMAux', mapIdxM.go, seqRight_eq, map_pure, seq_pure]
| Mathlib/Data/List/Indexes.lean | 238 | 242 |
/-
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.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# Partitions of rectangular b... |
@[simp]
theorem iUnion_filter_not (π : Prepartition I) (p : Box ι → Prop) :
(π.filter fun J => ¬p J).iUnion = π.iUnion \ (π.filter p).iUnion := by
simp only [Prepartition.iUnion]
convert
(@Set.biUnion_diff_biUnion_eq (ι → ℝ) (Box ι) π.boxes (π.filter p).boxes (↑) _).symm using 4
· simp +contextual
· rw... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 544 | 553 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 683 | 685 | |
/-
Copyright (c) 2021 Alex Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Zhao
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# Frobenius Number in Two Variables
... | have hmn : m + n ≤ m * n := add_le_mul hm hn
constructor
· push_neg
intro a b h
apply cop.mul_add_mul_ne_mul (add_one_ne_zero a) (add_one_ne_zero b)
simp only [Nat.sub_sub, smul_eq_mul] at h
zify [hmn] at h ⊢
rw [← sub_eq_zero] at h ⊢
rw [← h]
ring
· intro k hk
dsimp at hk
co... | Mathlib/NumberTheory/FrobeniusNumber.lean | 55 | 82 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | private def P : ℕ × ℤ × ℤ → Prop
| (r, s, t) => (r : ℤ) = x * s + y * t
| Mathlib/Data/Int/GCD.lean | 108 | 109 |
/-
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.Topology.Compactness.Compact
/-!
# Locally compact spaces
This file contains basic results about locally compact spac... |
theorem IsOpenQuotientMap.locallyCompactSpace [LocallyCompactSpace X] {f : X → Y}
(hf : IsOpenQuotientMap f) : LocallyCompactSpace Y where
local_compact_nhds := by
refine hf.surjective.forall.2 fun x U hU ↦ ?_
rcases local_compact_nhds (hf.continuous.continuousAt hU) with ⟨K, hKx, hKU, hKc⟩
| Mathlib/Topology/Compactness/LocallyCompact.lean | 172 | 177 |
/-
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, Yury Kudryashov
-/
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Logic.Function.Iterate
import Mathlib.Logic.Relation
/-!
# R... | · exact h i (by simp only [length_cons] at w; omega)
rintro ⟨h0, h⟩; constructor
· apply h0
simp
| Mathlib/Data/List/Chain.lean | 133 | 136 |
/-
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.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | mem_divisors.2 ⟨dvd_rfl, h⟩
theorem dvd_of_mem_divisors {m : ℕ} (h : n ∈ divisors m) : n ∣ m := by
cases m
| Mathlib/NumberTheory/Divisors.lean | 109 | 112 |
/-
Copyright (c) 2024 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
import Mathlib.Data.Finite.Prod
import Mathlib.MeasureTheory.SetSemiring
/-!
# Algebra of sets
In this file we define the notion of algebra of sets and give its bas... | /-- If a set belongs to the algebra of sets generated by `𝒜` then it can be written as a finite
union of finite intersections of sets which are in `𝒜` or have their complement in `𝒜`. -/
theorem mem_generateSetAlgebra_elim (s_mem : s ∈ generateSetAlgebra 𝒜) :
∃ A : Set (Set (Set α)), A.Finite ∧ (∀ a ∈ A, a.Fini... | Mathlib/MeasureTheory/SetAlgebra.lean | 172 | 212 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | refine le_lim (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩)
simp only [exp', const_apply, re_sum]
| Mathlib/Data/Complex/Exponential.lean | 239 | 240 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Enum
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
/-!
### Topology of ordinals
We prov... | simp_rw [exists_prop]
theorem mem_closed_iff_bsup (hs : IsClosed s) :
a ∈ s ↔
∃ (o : Ordinal) (_ho : o ≠ 0) (f : ∀ a < o, Ordinal),
| Mathlib/SetTheory/Ordinal/Topology.lean | 145 | 149 |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Path
/-!
# Trails and Eulerian trails
This module contains additional theory about trails, in... | /-- The edge set of an Eulerian graph is finite. -/
def IsEulerian.fintypeEdgeSet {u v : V} {p : G.Walk u v} (h : p.IsEulerian) :
Fintype G.edgeSet :=
Fintype.ofFinset h.isTrail.edgesFinset fun e => by
| Mathlib/Combinatorics/SimpleGraph/Trails.lean | 101 | 104 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Sites.LeftExact
import Mathlib.CategoryTheory.Sites.PreservesSheafification
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTh... |
lemma isLocallyInjective_iff_of_fac {φψ : F₁ ⟶ F₃} (fac : φ ≫ ψ = φψ) [IsLocallyInjective J ψ] :
IsLocallyInjective J φψ ↔ IsLocallyInjective J φ := by
constructor
· intro
exact isLocallyInjective_of_isLocallyInjective_fac J fac
· intro
rw [← fac]
| Mathlib/CategoryTheory/Sites/LocallyInjective.lean | 146 | 153 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... |
end LieSubalgebra
| Mathlib/Algebra/Lie/Subalgebra.lean | 697 | 699 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | theorem mul_snd (x y : X q) : (x * y).2 = x.1 * y.2 + x.2 * y.1 :=
rfl
| Mathlib/NumberTheory/LucasLehmer.lean | 241 | 242 |
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | | pair cf cg pf pg => exact pf.pair pg
| comp cf cg pf pg => exact pf.comp pg
| prec cf cg pf pg => exact pf.prec pg
| rfind' cf pf => exact pf.rfind'
| Mathlib/Computability/PartrecCode.lean | 550 | 554 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | cons x fun j ↦ _root_.cast (congr_arg α (congr_fun succAbove_zero j)) (p j) := by
refine insertNth_eq_iff.2 ⟨by simp, ?_⟩
ext j
convert (cons_succ x p j).symm
| Mathlib/Data/Fin/Tuple/Basic.lean | 871 | 875 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | MapsTo (fun x ↦ (f₁ x.1, f₂ x.2)) (s₁ ×ˢ s₂) (t₁ ×ˢ t₂) :=
fun _x hx ↦ ⟨h₁ hx.1, h₂ hx.2⟩
lemma BijOn.prodMap (h₁ : BijOn f₁ s₁ t₁) (h₂ : BijOn f₂ s₂ t₂) :
BijOn (fun x ↦ (f₁ x.1, f₂ x.2)) (s₁ ×ˢ s₂) (t₁ ×ˢ t₂) :=
⟨h₁.mapsTo.prodMap h₂.mapsTo, h₁.injOn.prodMap h₂.injOn, h₁.surjOn.prodMap h₂.surjOn⟩
| Mathlib/Data/Set/Function.lean | 1,212 | 1,217 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | liminf f v = sSup (⋃ (j : Subtype p), ⋂ (i : s j), Iic (f i)) := by
| Mathlib/Order/LiminfLimsup.lean | 276 | 276 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Defs
/-!
# Equitable functions
This file defines eq... |
end Set
open Set
| Mathlib/Data/Set/Equitable.lean | 80 | 83 |
/-
Copyright (c) 2023 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Topology.Order.LowerUpperTopology
import Mathlib.Topology.Order.ScottTopology
/-!
# Lawson topology
This file introduces the Lawson topology... | rw [@IsScott.topology_eq α univ _ S _, @IsLawson.topology_eq_lawson α _ L _]
exact inf_le_right
| Mathlib/Topology/Order/LawsonTopology.lean | 187 | 189 |
/-
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.Data.Fintype.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | @[simp]
theorem nil_toFinset : (@nil α).toFinset = ∅ :=
| Mathlib/Data/List/Cycle.lean | 703 | 704 |
/-
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.IsManifold.ExtChartAt
import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# `C^n` functions betwee... | exact forall₂_congr fun x hx => contMDiffWithinAt_iff_image he he' hs (hs hx) (h2s hx)
theorem contMDiffOn_iff_of_mem_maximalAtlas' (he : e ∈ maximalAtlas I n M)
(he' : e' ∈ maximalAtlas I' n M') (hs : s ⊆ e.source) (h2s : MapsTo f s e'.source) :
ContMDiffOn I I' n f s ↔
| Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 466 | 470 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.IntegralCompProd
import Mathlib.Probability.Kernel.Disintegration.StandardBorel
/-!
# Lebesgue and Bochner integrals of con... | conv_rhs => rw [← κ.disintegrate κ.condKernel]
rw [Kernel.lintegral_compProd _ _ _ hf]
lemma setLIntegral_condKernel (hf : Measurable f) (a : α) {s : Set β}
| Mathlib/Probability/Kernel/Disintegration/Integral.lean | 61 | 64 |
/-
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.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | @[deprecated (since := "2025-01-21")] alias condexpIndL1_add := condExpIndL1_add
theorem condExpIndL1_smul (c : ℝ) (x : G) :
condExpIndL1 hm μ s (c • x) = c • condExpIndL1 hm μ s x := by
by_cases hs : MeasurableSet s
swap; · simp_rw [condExpIndL1_of_not_measurableSet hs]; rw [smul_zero]
by_cases hμs : μ s = ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 220 | 227 |
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