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 |
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/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... |
variable [IsFractionRing R₁ K] [IsDomain R₁]
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 350 | 351 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Partition.Equipartition
/-!
# Turán's theorem
In this file we prove Turán's theorem, the first importan... | use G ⊔ edge x y, inferInstance, cf.sup_edge x y
convert Nat.lt.base #G.edgeFinset
convert G.card_edgeFinset_sup_edge _ hn
rwa [h.not_adj_iff_part_eq]
/-- **Turán's theorem**, forward direction.
Any `r + 1`-cliquefree Turán-maximal graph on `n` vertices is isomorphic to `turanGraph n r`. -/
theorem nonempty_i... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 258 | 270 |
/-
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.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | rw [Pi.add_apply, exp_add]
ring
simp_rw [this, div_eq_mul_inv]
rw [integral_mul_const]
lemma tilted_tilted (hf : Integrable (fun x ↦ exp (f x)) μ) (g : α → ℝ) :
(μ.tilted f).tilted g = μ.tilted (f + g) := by
cases eq_zero_or_neZero μ with
| inl h => simp [h]
| inr h0 =>
ext1 s hs
... | Mathlib/MeasureTheory/Measure/Tilted.lean | 242 | 255 |
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.IsPrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Multiplicity
import Mathlib.Data.Z... | let c₂ : Fin (s + 1) → Associates N := fun t => d ⟨c₁ t, le_trans (hc₁''.2 ⟨t, by simp⟩) hs'⟩
have c₂_def : ∀ t, c₂ t = d ⟨c₁ t, _⟩ := fun t => rfl
rw [← c₂_def]
refine (eq_pow_second_of_chain_of_has_chain hs (fun t u h => ?_)
(@fun r => ⟨@fun hr => ?_, ?_⟩) ?_).symm
· rw [c₂_def, c₂_def, Subtype.coe_lt_c... | Mathlib/RingTheory/ChainOfDivisors.lean | 252 | 281 |
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Topology.MetricS... | · push_neg at h
refine Or.inr fun s hs1 hs2 => isOpen_iff_mem_nhds.mpr ?_
rintro z ⟨w, hw1, rfl⟩
exact (hg w (hs1 hw1)).eventually_constant_or_nhds_le_map_nhds.resolve_left (h w (hs1 hw1))
(image_mem_map (hs2.mem_nhds hw1))
| Mathlib/Analysis/Complex/OpenMapping.lean | 163 | 172 |
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller, Pim Otte
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph
import Mathlib.Com... | ⟨s(v, (h v.property).choose), (h v.property).choose_spec.1⟩
theorem IsMatching.toEdge_eq_of_adj (h : M.IsMatching) (hv : v ∈ M.verts) (hvw : M.Adj v w) :
h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ := by
simp only [IsMatching.toEdge, Subtype.mk_eq_mk]
| Mathlib/Combinatorics/SimpleGraph/Matching.lean | 70 | 74 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,758 | 1,760 | |
/-
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.Data.Set.Finite.Lattice
/-!
# Partitions based on membership of a sequence of sets
Let `f : ℕ → Set α` be a sequence of sets. For `n : ℕ`, we can form th... | lemma memPartitionSet_succ (f : ℕ → Set α) (n : ℕ) (a : α) [Decidable (a ∈ f n)] :
memPartitionSet f (n + 1) a
= if a ∈ f n then memPartitionSet f n a ∩ f n else memPartitionSet f n a \ f n := by
simp [memPartitionSet]
| Mathlib/Data/Set/MemPartition.lean | 117 | 121 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... |
theorem smeval_X_mul : (X * p).smeval x = x * p.smeval x := by
induction p using Polynomial.induction_on' with
| add p q ph qh => simp only [smeval_add, ph, qh, mul_add]
| monomial n a =>
rw [← monomial_one_one_eq_X, monomial_mul_monomial, smeval_monomial, one_mul, npow_add,
npow_one, ← mul_smul_comm... | Mathlib/Algebra/Polynomial/Smeval.lean | 210 | 216 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | section FastJacobi
/-!
### Fast computation of the Jacobi symbol
We follow the implementation as in `Mathlib.Tactic.NormNum.LegendreSymbol`.
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 486 | 490 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... | rcases eq_or_ne x y with (rfl | h)
· exact differentiableWithinAt_insert_self
apply differentiableWithinAt_congr_nhds
exact nhdsWithin_insert_of_ne h
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 620 | 623 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplexBiprod
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
/-! The homotopy ... | @[reassoc (attr := simp)]
lemma inrX_desc_f (i : ι) :
inrX φ i ≫ (desc φ α hα).f i = α.f i := by
dsimp [desc]
split_ifs <;> simp
| Mathlib/Algebra/Homology/HomotopyCofiber.lean | 310 | 314 |
/-
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.Algebra.Pi
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndepe... | Mathlib/LinearAlgebra/StdBasis.lean | 278 | 280 | |
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, David Loeffler
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Numbe... | have hnp1 : IsUnit ((n + 1)! : ℚ) := IsUnit.mk0 _ (mod_cast factorial_ne_zero (n + 1))
rw [← (hnp1.map (algebraMap ℚ A)).mul_right_inj]
-- do trivial rearrangements to make RHS (n+1)*t^n
rw [mul_left_comm, ← RingHom.map_mul]
change _ = t ^ n * algebraMap ℚ A (((n + 1) * n ! : ℕ) * (1 / n !))
rw [cast_mul, m... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 221 | 259 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 1,368 | 1,371 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... | · rw [disjUnion_tag_of_mem_right _ hJ]
@[simp]
theorem integralSum_add (f g : ℝⁿ → E) (vol : ι →ᵇᵃ E →L[ℝ] F) (π : TaggedPrepartition I) :
integralSum (f + g) vol π = integralSum f vol π + integralSum g vol π := by
simp only [integralSum, Pi.add_apply, (vol _).map_add, Finset.sum_add_distrib]
| Mathlib/Analysis/BoxIntegral/Basic.lean | 127 | 133 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.GCongr.CoreAttrs
import Mathlib.Tactic.Common
import Mathlib.Tactic.... | refine ⟨fun h => ?_, congr_arg _⟩
obtain hnm | rfl | hnm := lt_trichotomy n m
· rw [← factorial_lt <| lt_of_succ_lt hn, h] at hnm
cases lt_irrefl _ hnm
· rfl
rw [← one_lt_factorial, h, one_lt_factorial] at hn
| Mathlib/Data/Nat/Factorial/Basic.lean | 113 | 118 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | end PosNum
| Mathlib/Data/Num/Lemmas.lean | 711 | 711 |
/-
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.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.OpenCover
import Mathlib.Topology.LocallyClosed
/-!
# Properties of maps that are local at the targe... | MapsTo.coe_restrict, restrict_eq, ← Filter.comap_comap]
constructor
· intro H i x
rw [Function.comp_apply, ← H, ← IsInducing.subtypeVal.nhds_eq_comap]
· intro H x
obtain ⟨i, hi⟩ := Opens.mem_iSup.mp (show f x ∈ iSup U by simp [hU.iSup_eq_top])
simpa [← ((h.1 _ (U i).2).isOpenEmbedding_subtypeVal).... | Mathlib/Topology/LocalAtTarget.lean | 148 | 156 |
/-
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.Data.Bool.Basic
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.CoreAttrs
/-!
# (Semi-)lattices
Semilatti... | @SemilatticeInf.toPartialOrder _ (SemilatticeInf.mk' inf_comm inf_assoc inf_idem) :=
| Mathlib/Order/Lattice.lean | 478 | 478 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... |
@[simp]
theorem kahler_rightAngleRotation_left (x y : E) :
o.kahler (J x) y = -Complex.I * o.kahler x y := by
simp only [o.areaForm_rightAngleRotation_left, o.inner_rightAngleRotation_left,
o.kahler_apply_apply, Complex.ofReal_neg, Complex.real_smul]
linear_combination ω x y * Complex.I_sq
@[simp]
theorem... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 423 | 438 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 667 | 670 | |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | Quot.liftOn c F
(by
suffices ∀ {α β}, α ≈ β → F α ≤ F β from
fun α β h => (this h).antisymm (this (Setoid.symm h))
rintro α β ⟨f⟩
refine le_ciInf_iff'.2 fun i => ?_
haveI := @RelEmbedding.isWellOrder _ _ (f ⁻¹'o i.1) _ (↑(RelIso.preimage f i.1)) i.2
exact
(ciInf_le' _... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,010 | 1,035 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Set.UnorderedInterval
/-!
# Locally... | Mathlib/Order/Interval/Finset/Defs.lean | 1,350 | 1,351 | |
/-
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... | · rintro ⟨f⟩
have le_one := finrank_hom_simple_simple_le_one 𝕜 X Y
have zero_lt : 0 < finrank 𝕜 (X ⟶ Y) :=
finrank_pos_iff_exists_ne_zero.mpr ⟨f.hom, (isIso_iff_nonzero f.hom).mp inferInstance⟩
omega
theorem finrank_hom_simple_simple_eq_zero_iff (X Y : C) [FiniteDimensional 𝕜 (X ⟶ X)]
[Finit... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 177 | 190 |
/-
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.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | Mathlib/Topology/MetricSpace/Isometry.lean | 631 | 633 | |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | theorem one_le_finprod' {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedMonoid M]
{f : α → M} (hf : ∀ i, 1 ≤ f i) :
1 ≤ ∏ᶠ i, f i :=
finprod_induction _ le_rfl (fun _ _ => one_le_mul) hf
| Mathlib/Algebra/BigOperators/Finprod.lean | 258 | 261 |
/-
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
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Mathlib/Analysis/Calculus/MeanValue.lean | 1,050 | 1,055 | |
/-
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.Localization.CalculusOfFractions.Fractions
import Mathlib.CategoryTheory.Localization.HasLocalization
import Mathlib.CategoryTheory.Preadditive.Ad... | attribute [irreducible] preadditive
include W in
lemma functor_additive_iff {E : Type*} [Category E] [Preadditive E] [Preadditive D] [L.Additive]
(G : D ⥤ E) :
G.Additive ↔ (L ⋙ G).Additive := by
constructor
· intro
infer_instance
· intro h
suffices ∀ ⦃X Y : C⦄ (f g : L.obj X ⟶ L.obj Y), G.map (f ... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 320 | 342 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | theorem Ico_filter_le (a b c : α) : ((Ico a b).filter fun x => c ≤ x) = Ico (max a c) b := by
rw [Ico, Ico, ← Finset.filter_val, Finset.Ico_filter_le]
@[simp]
| Mathlib/Order/Interval/Multiset.lean | 287 | 290 |
/-
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.LinearMap
import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
import Mathlib.MeasureTheory.Function.StronglyMeasur... |
@[deprecated (since := "2025-02-21")]
alias Memℒp.const_inner := MemLp.const_inner
| Mathlib/MeasureTheory/Function/L2Space.lean | 74 | 76 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | Fintype (G₁ ⊓ G₂).edgeSet := by
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 544 | 544 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | let ⟨n, hn⟩ := hf.exists_zpow_eq hx hy
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 284 | 284 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... | simpa [pow_two, mul_assoc] using T_add_two R 0
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 113 | 114 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Combinatorics.SimpleGraph.Regularity.Bound
import Mathlib.Combinatorics.SimpleGraph.Regularity.Equitabilise
import Mathlib.Comb... | card_le_m_add_one_of_mem_chunk_parts <| star_subset_chunk hs
| Mathlib/Combinatorics/SimpleGraph/Regularity/Chunk.lean | 194 | 195 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,388 | 1,392 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib... | (q.hasStrictDerivAt_aeval x).hasDerivAt
protected theorem hasDerivWithinAt (x : 𝕜) (s : Set 𝕜) :
HasDerivWithinAt (fun x => p.eval x) (p.derivative.eval x) s x :=
| Mathlib/Analysis/Calculus/Deriv/Polynomial.lean | 66 | 69 |
/-
Copyright (c) 2022 Mario Carneiro, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Heather Macbeth, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.PosPart
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Int.CharZero
im... | rw [max_def]; split_ifs <;> assumption
| Mathlib/Tactic/Positivity/Basic.lean | 98 | 99 |
/-
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.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | -/
theorem Isometry.dimH_image (hf : Isometry f) (s : Set X) : dimH (f '' s) = dimH s :=
le_antisymm (hf.lipschitz.dimH_image_le _) (hf.antilipschitz.le_dimH_image _)
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 371 | 376 |
/-
Copyright (c) 2019 Minchao Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Minchao Wu, Chris Hughes, Mantas Bakšys
-/
import Mathlib.Data.List.Basic
import Mathlib.Order.BoundedOrder.Lattice
import Mathlib.Data.List.Induction
import Mathlib.Order.MinMax
import Ma... |
/-- The minimum value in a non-empty `List`. -/
def minimum_of_length_pos (h : 0 < l.length) : α :=
maximum_of_length_pos (α := αᵒᵈ) h
| Mathlib/Data/List/MinMax.lean | 399 | 403 |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.List.ProdSigma
import Mathlib.Data.List.Pi
/-!
Type class for finitely enumerabl... |
@[simp]
theorem Finset.mem_enum [DecidableEq α] (s : Finset α) (xs : List α) :
s ∈ Finset.enum xs ↔ ∀ x ∈ s, x ∈ xs := by
induction xs generalizing s with
| nil => simp [enum, eq_empty_iff_forall_not_mem]
| cons x xs ih =>
simp only [enum, List.bind_eq_flatMap, List.mem_flatMap, List.mem_cons, List.mem... | Mathlib/Data/FinEnum.lean | 132 | 163 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | end Stream'
| Mathlib/Data/Stream/Init.lean | 723 | 729 |
/-
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.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | map_one' := coeff_one_zero
map_mul' := mul_coeff_zero
map_zero' := coeff_zero 0
map_add' p q := coeff_add p q 0
| Mathlib/Algebra/Polynomial/Coeff.lean | 127 | 131 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... |
private lemma ghostFun_nsmul (m : ℕ) (x : WittVector p R) : ghostFun (m • x) = m • ghostFun x := by
| Mathlib/RingTheory/WittVector/Basic.lean | 190 | 191 |
/-
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.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | Mathlib/RingTheory/Polynomial/Basic.lean | 457 | 457 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Bhavik Mehta, Stuart Presnell
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Binomial coefficients
This file defines binomial coeffic... |
theorem choose_eq_zero_iff {n k : ℕ} : n.choose k = 0 ↔ n < k :=
⟨fun h => lt_of_not_ge (mt Nat.choose_pos h.symm.not_lt), Nat.choose_eq_zero_of_lt⟩
theorem succ_mul_choose_eq : ∀ n k, succ n * choose n k = choose (succ n) (succ k) * succ k
| 0, 0 => by decide
| 0, k + 1 => by simp [choose]
| Mathlib/Data/Nat/Choose/Basic.lean | 116 | 122 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
/-!
# Construction of projections for the Dold-Kan correspondence
In this file... | simp only [hnaq, Nat.succ_eq_add_one, add_assoc]
rfl)
simp only [Fin.succ_mk] at eq
simp only [eq, zero_comp]
| Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 109 | 113 |
/-
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 | 1,324 | 1,324 | |
/-
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... | show f (r • n) ∈ I • N.map f from
(f.map_smul r n).symm ▸ smul_mem_smul hr (mem_map_of_mem hn)) <|
| Mathlib/RingTheory/Ideal/Operations.lean | 99 | 100 |
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
import Mathlib.Algebra.Module.Submodul... | variable [Module R M] [Module R₂ M₂]
variable (p : Submodule R M)
variable {τ₁₂ : R →+* R₂}
| Mathlib/Algebra/Module/Submodule/Range.lean | 268 | 270 |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Retract
/-!
# Lifting properties
This file defines the lifting property of two morph... | simp only [← h.i_w_assoc, sq'.fac_left, h.retract_left_assoc,
Arrow.mk_left, Category.id_comp]}⟩⟩
lemma RetractArrow.rightLiftingProperty
| Mathlib/CategoryTheory/LiftingProperties/Basic.lean | 141 | 144 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | simp
/-- Composing a functor with both functors of an equivalence yields a naturally isomorphic
functor. -/
| Mathlib/CategoryTheory/Equivalence.lean | 324 | 327 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Complex.Cardinality
import Mathlib.Data.Complex.Modu... | as additive groups. -/
theorem Complex.nonempty_linearEquiv_real : Nonempty (ℂ ≃ₗ[ℚ] ℝ) :=
LinearEquiv.nonempty_equiv_iff_rank_eq.mpr <| by simp
| Mathlib/Data/Complex/FiniteDimensional.lean | 73 | 76 |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,400 | 1,405 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | Mathlib/Data/Seq/Computation.lean | 1,265 | 1,267 | |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# L'Hôpital's rule for 0/0 indeterminate forms
In this file, we ... | exact HasDerivAt.lhopital_zero_atTop hdf' hdg' hg' hftop hgtop hdiv
/-- **L'Hôpital's rule** for approaching -∞, `deriv` version -/
theorem lhopital_zero_atBot (hdf : ∀ᶠ x : ℝ in atBot, DifferentiableAt ℝ f x)
(hg' : ∀ᶠ x : ℝ in atBot, deriv g x ≠ 0) (hfbot : Tendsto f atBot (𝓝 0))
(hgbot : Tendsto g atBot ... | Mathlib/Analysis/Calculus/LHopital.lean | 434 | 445 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 645 | 646 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Data.ENNReal.Action
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.MetricSpace.Algebra
... |
variable [NormedDivisionRing α] [SeminormedAddGroup β]
variable [MulActionWithZero α β] [IsBoundedSMul α β]
theorem norm_smul (r : α) (x : β) : ‖r • x‖ = ‖r‖ * ‖x‖ := by
by_cases h : r = 0
· simp [h, zero_smul α x]
· refine le_antisymm (norm_smul_le r x) ?_
| Mathlib/Analysis/Normed/MulAction.lean | 90 | 97 |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FDRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
/-!
# Characters of representations
Th... | theorem char_tensor (V W : FDRep k G) : (V ⊗ W).character = V.character * W.character := by
ext g; convert trace_tensorProduct' (V.ρ g) (W.ρ g)
| Mathlib/RepresentationTheory/Character.lean | 59 | 60 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCa... | LocallyConstant.ofIsClopen_fiber_zero]
exact h
open Classical in
theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 116 | 126 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.RelIso.Set
import Mathlib.Order.WellQuasiOrder
import Mathlib.Tactic.TFAE
/-!
# Well-... | exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|
ha.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)
section IsRefl
variable [IsRefl α r]
protected theorem Finite.partiallyWellOrderedOn (hs : s.Finite) : s.PartiallyWellOrderedOn r :=
| Mathlib/Order/WellFoundedSet.lean | 304 | 310 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | @[reassoc (attr := simp)]
theorem equalizerSubobject_arrow' :
(equalizerSubobjectIso f g).inv ≫ (equalizerSubobject f g).arrow = equalizer.ι f g := by
| Mathlib/CategoryTheory/Subobject/Limits.lean | 50 | 52 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 701 | 701 | |
/-
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.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullbac... |
variable [LocallySmall.{t} C] [Small.{t} I]
instance : Small.{t} (FunctorObjIndex f πX) := by
let φ (x : FunctorObjIndex f πX) :
| Mathlib/CategoryTheory/SmallObject/Construction.lean | 159 | 163 |
/-
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.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | -- But this is an elementary calculation.
constructor
· rintro ⟨y, hyU, hyx⟩
apply Set.mem_iUnion_of_mem (x⁻¹ * y)
simp_all
· rintro ⟨_, ⟨k, rfl⟩, _, ⟨(hk : φ k = 1), rfl⟩, hx⟩
use x * k, hx
rw [map_mul, hk, mul_one]
| Mathlib/Topology/Algebra/Group/Basic.lean | 804 | 812 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,245 | 2,252 | |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | theorem dist_add_add_right (n k m : ℕ) : dist (n + k) (m + k) = dist n m :=
| Mathlib/Data/Nat/Dist.lean | 57 | 57 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
/-!
# Angles between vectors
This fil... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 338 | 342 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | Mathlib/Tactic/Ring/Basic.lean | 357 | 357 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.Defs
import Mathlib.Data.DFinsupp.Module
/-!
# Pointwise order on finitely supported dependent functions
This file lifts order struc... | theorem support_tsub : (f - g).support ⊆ f.support := by
simp +contextual only [subset_iff, tsub_eq_zero_iff_le, mem_support_iff,
Ne, coe_tsub, Pi.sub_apply, not_imp_not, zero_le, imp_true_iff]
| Mathlib/Data/DFinsupp/Order.lean | 265 | 267 |
/-
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... |
@[deprecated (since := "2024-10-24")]
| Mathlib/Data/Nat/Factorization/Basic.lean | 331 | 332 |
/-
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.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... |
variable {k n : ℕ}
/-- The Roth number of a natural `N` is the largest integer `m` for which there is a subset of
`range N` of size `m` with no arithmetic progression of length 3.
Trivially, `rothNumberNat N ≤ N`, but Roth's theorem (proved in 1953) shows that
`rothNumberNat N = o(N)` and the construction by Behrend ... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 393 | 405 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
/-!
# Wide pullbacks
We define the category `WidePullbackShape`, (re... |
/-- The obvious functor `(WidePullbackShape J)ᵒᵖ ⥤ WidePushoutShape J` -/
| Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 434 | 435 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw
obtain ⟨y, hy⟩ := hzw
rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul]
| Mathlib/RingTheory/IsAdjoinRoot.lean | 186 | 188 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | · rintro ⟨r, hr, rfl⟩
rw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right hx]
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 302 | 303 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Down-compressions
This file defines down-compression.
Down-compressing `𝒜 : Finset (Fins... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 283 | 290 | |
/-
Copyright (c) 2023 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.Algebra.Group.Subgroup.Lattice
/-!
# Submonoid of un... | @[to_additive]
lemma mem_units_iff_val_mem (H : Subgroup G) (x : Gˣ) : x ∈ H.units ↔ (x : G) ∈ H := by
simp_rw [Submonoid.mem_units_iff, mem_toSubmonoid, val_inv_eq_inv_val, inv_mem_iff, and_self]
| Mathlib/Algebra/Group/Submonoid/Units.lean | 310 | 312 |
/-
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... | rw [take]
cases s.destruct with
| none => simp
| some v =>
obtain ⟨x, r⟩ := v
simpa using ih
theorem length_take_of_le_length {s : Seq α} {n : ℕ}
(hle : ∀ h : s.Terminates, n ≤ s.length h) : (s.take n).length = n := by
induction n generalizing s with
| zero => simp [take]
| succ n... | Mathlib/Data/Seq/Seq.lean | 782 | 802 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.Algebra.Group.Submonoid.Finsupp
import Mathlib.Order.Filter.AtTopBot.Defs
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Gra... | `𝒜 i` contains both numerator and denominator. -/
def deg (f : HomogeneousLocalization 𝒜 x) : ι :=
(Quotient.out f).deg
| Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 498 | 500 |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Dynamics.PeriodicPts.Lemmas
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.GroupAction.Basic
/-!
# Period of a group action
This modul... | theorem period_dvd_orderOf (m : M) (a : α) : period m a ∣ orderOf m := by
rw [← pow_smul_eq_iff_period_dvd, pow_orderOf_eq_one, one_smul]
| Mathlib/GroupTheory/GroupAction/Period.lean | 87 | 88 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | simp only [Nodup, pairwise_append, disjoint_iff_ne]
theorem disjoint_of_nodup_append {l₁ l₂ : List α} (d : Nodup (l₁ ++ l₂)) : Disjoint l₁ l₂ :=
| Mathlib/Data/List/Nodup.lean | 179 | 181 |
/-
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... | · have : i < length (ofFn f) := by rwa [length_ofFn]
rw [prod_take_succ _ _ this]
have A : ({j | j.val < i + 1} : Finset (Fin n)) =
insert ⟨i, h⟩ ({j | Fin.val j < i} : Finset (Fin n)) := by
ext ⟨_, _⟩
simp [Nat.lt_succ_iff_lt_or_eq, or_comm]
rw [A, prod_insert (by simp),... | Mathlib/Algebra/BigOperators/Fin.lean | 441 | 447 |
/-
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.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Probability.Kernel.MeasurableLIntegral
/-!
# Measurability of the integral against a kernel
Th... | (hf : StronglyMeasurable (uncurry f)) : StronglyMeasurable fun y => ∫ x, f x y ∂κ y :=
(hf.comp_measurable measurable_swap).integral_kernel_prod_right'
theorem StronglyMeasurable.integral_kernel_prod_left' ⦃f : β × α → E⦄ (hf : StronglyMeasurable f) :
StronglyMeasurable fun y => ∫ x, f (x, y) ∂κ y :=
(hf.c... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 136 | 145 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | iterate_derivative_C h
@[simp]
| Mathlib/Algebra/Polynomial/Derivative.lean | 222 | 224 |
/-
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.Convex.Topology
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.SpecificLimits.Basi... | apply tendsto_inv_nhdsGT_zero.comp
simp only [nhdsWithin, tendsto_inf, tendsto_principal, mem_Ioi, norm_pos_iff, ne_eq,
| Mathlib/Analysis/Calculus/TangentCone.lean | 393 | 394 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Set.Image
/-!
# Support of a func... | mulSupport (update f x y) = insert x (mulSupport f) := by
ext a; rcases eq_or_ne a x with rfl | hne <;> simp [*]
@[to_additive]
| Mathlib/Algebra/Group/Support.lean | 73 | 76 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | (p.1 ^ p.2 * log p.1) • ContinuousLinearMap.snd ℂ ℂ ℂ) p := by
have A : p.1 ≠ 0 := slitPlane_ne_zero hp
have : (fun x : ℂ × ℂ => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) :=
((isOpen_ne.preimage continuous_fst).eventually_mem A).mono fun p hp =>
cpow_def_of_ne_zero hp _
rw [cpow_sub _ _ A... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 31 | 42 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Path
/-!
# Path connectedness
Continuing from `Mathlib.Topology.Path`, this file defines path components and path-connected
spaces.
## Main... | Mathlib/Topology/Connected/PathConnected.lean | 730 | 742 | |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` w... | theorem coe_bot : ↑(⊥ : ConvexCone 𝕜 E) = (∅ : Set E) :=
rfl
| Mathlib/Analysis/Convex/Cone/Basic.lean | 141 | 142 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | replace hζ : IsPrimitiveRoot ζ (p ^ (0 + 1)) := by simpa using hζ
exact hζ.finite_quotient_span_sub_one
/-- In a `p ^ (k + 1)`-th cyclotomic extension of `ℚ`, we have that
`ζ - 1` divides `p` in `𝓞 K`. -/
lemma toInteger_sub_one_dvd_prime [hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRo... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 507 | 513 |
/-
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 | 1,657 | 1,670 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | lemma trace_lie {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M] (f g : Module.End R M) :
trace R M ⁅f, g⁆ = 0 := by
rw [Ring.lie_def, map_sub, trace_mul_comm]
| Mathlib/LinearAlgebra/Trace.lean | 110 | 112 |
/-
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.Topology.UniformSpace.AbstractCompletion
/-!
# Hausdorff completions of uniform spaces
The goal is to construct a left-adjoint to the... | rw [lift'_lift'_assoc]
· exact monotone_id.compRel monotone_id
· exact monotone_gen
_ ≤ (𝓤 α).lift' gen := lift'_mono comp_le_uniformity le_rfl
instance : UniformSpace (CauchyFilter α) :=
UniformSpace.ofCore
{ uniformity := (𝓤 α).lift' gen
refl := principal_le_lift'.2 fun _s hs ⟨a, ... | Mathlib/Topology/UniformSpace/Completion.lean | 116 | 129 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... | · rw [h, leadingCoeff_zero, coeff_zero]
· rw [leadingCoeff_of_ne h, order_of_ne h]
| Mathlib/RingTheory/HahnSeries/Basic.lean | 395 | 397 |
/-
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.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | Mathlib/Topology/MetricSpace/Bounded.lean | 629 | 633 | |
/-
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 | 1,508 | 1,512 | |
/-
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... | theorem iUnion_singleton_eq_range (f : α → β) : ⋃ x : α, {f x} = range f := by
ext x
| Mathlib/Data/Set/Lattice.lean | 991 | 992 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lorenzo Luccioli, Rémy Degenne, Alexander Bentkamp
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform
import Mathlib.Probability.Moments.ComplexMGF
/-!
# Gaussian dis... | /-- The gaussian distribution pdf integrates to 1 when the variance is not zero. -/
lemma lintegral_gaussianPDFReal_eq_one (μ : ℝ) {v : ℝ≥0} (h : v ≠ 0) :
∫⁻ x, ENNReal.ofReal (gaussianPDFReal μ v x) = 1 := by
rw [← ENNReal.toReal_eq_one_iff]
have hfm : AEStronglyMeasurable (gaussianPDFReal μ v) volume :=
(... | Mathlib/Probability/Distributions/Gaussian.lean | 96 | 113 |
/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam, Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Algebra.Spectrum.Basic
/-!
# Unitar... | Mathlib/Algebra/Star/Unitary.lean | 255 | 259 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... |
theorem range_coe (f : M →[L] N) : (range f : Set N) = Set.range f :=
rfl
@[simp]
theorem mem_range {f : M →[L] N} {x} : x ∈ range f ↔ ∃ y, f y = x :=
Iff.rfl
| Mathlib/ModelTheory/Substructures.lean | 788 | 795 |
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