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 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 Sbtw.left_mem_image_Ioi {x y z : P} (h : Sbtw R x y z) :
x ∈ lineMap z y '' Set.Ioi (1 : R) :=
h.symm.right_mem_image_Ioi
| Mathlib/Analysis/Convex/Between.lean | 688 | 691 |
/-
Copyright (c) 2019 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.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | there exists an object `S`, with a morphism `T X : S ⟶ X` from each `X`,
such that the triangles commute: `T X ≫ f = T Y`, for `f : X ⟶ Y` in the `Finset`.
-/
theorem inf_exists :
| Mathlib/CategoryTheory/Filtered/Basic.lean | 678 | 681 |
/-
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... | theorem setToL1S_zero_left (f : α →₁ₛ[μ] E) : setToL1S (0 : Set α → E →L[ℝ] F) f = 0 :=
SimpleFunc.setToSimpleFunc_zero _
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 112 | 114 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
/-!
# Basic facts about real ... | Subset.antisymm (range_subset_iff.2 norm_nonneg) fun _ => exists_norm_eq E
theorem nnnorm_surjective : Surjective (nnnorm : E → ℝ≥0) := fun c =>
(exists_norm_eq E c.coe_nonneg).imp fun _ h => NNReal.eq h
| Mathlib/Analysis/NormedSpace/Real.lean | 124 | 128 |
/-
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.Compactness.Bases
import Mathlib.Topology.NoetherianSpace
/-!
# Quasi-separated spaces
A topological space is quasi-separated if the intersections... | rw [← Set.preimage_inter, Set.image_preimage_eq_inter_range, Set.inter_eq_left]
exact Set.inter_subset_left.trans (hU.trans (Set.image_subset_range _ _))
· intro x hx
rw [← h.injective.injOn.mem_image_iff (Set.subset_univ _) trivial]
exact hU hx
· rw [h.isCompact_iff]
convert hU''
rw [Set.im... | Mathlib/Topology/QuasiSeparated.lean | 64 | 86 |
/-
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... | have := (atTop_neBot_iff.1 h).2
rw [measure_iUnion_eq_iSup_accumulate]
exact tendsto_atTop_iSup fun i j hij ↦ by gcongr
/-- Continuity from above: the measure of the intersection of a decreasing sequence of measurable
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 591 | 595 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 834 | 842 | |
/-
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... |
@[simp]
| Mathlib/Data/Nat/Choose/Basic.lean | 88 | 89 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | Bijective (· ^ n : G → G) :=
(powCoprime hn).bijective
/- TODO: Generalise to `Submonoid.powers`. -/
@[to_additive]
theorem image_range_orderOf [DecidableEq G] :
| Mathlib/GroupTheory/OrderOfElement.lean | 968 | 973 |
/-
Copyright (c) 2023 Yaël Dillies, Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Christopher Hoskin
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Or... |
lemma supClosed_range [FunLike F α β] [SupHomClass F α β] (f : F) : SupClosed (Set.range f) := by
| Mathlib/Order/SupClosed.lean | 72 | 73 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,027 | 1,038 | |
/-
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.ExactSequence
import Mathlib.Algebra.Homology.ShortComplex.Limits
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The snake lemma
The ... | @[reassoc (attr := simp)] lemma w₁₃_τ₁ : S.v₁₂.τ₁ ≫ S.v₂₃.τ₁ = 0 := by
rw [← comp_τ₁, S.w₁₃, zero_τ₁]
| Mathlib/Algebra/Homology/ShortComplex/SnakeLemma.lean | 124 | 125 |
/-
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... |
variable [CompleteLattice β]
| Mathlib/Data/Set/Lattice.lean | 1,360 | 1,361 |
/-
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.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | · by_contra h''
rw [h'] at h''
simp only [pow_zero, one_mul, coeff_one, one_ne_zero, ↓reduceIte, zero_mul, add_zero,
CharP.cast_eq_zero, zero_add, mul_one, not_true_eq_false] at h''
norm_num at h''
· rw [ih]
· conv => lhs; arg 2; rw [mul_comm, ← mul_as... | Mathlib/RingTheory/PowerSeries/Basic.lean | 682 | 693 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | Mathlib/Probability/Density.lean | 332 | 342 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | theorem toOrderedSMul : OrderedSMul ℝ K :=
OrderedSMul.mk' fun a b r hab hr => by
| Mathlib/Analysis/RCLike/Basic.lean | 859 | 860 |
/-
Copyright (c) 2024 Frédéric Marbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Marbach
-/
import Mathlib.Algebra.Lie.Derivation.AdjointAction
import Mathlib.Algebra.Lie.Killing
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Derivations of ... | lemma killingForm_restrict_range_ad [Module.Finite R L] :
(killingForm R 𝔻).restrict 𝕀 = killingForm R 𝕀 := by
rw [← (ad_isIdealMorphism R L).eq, ← LieIdeal.killingForm_eq]
| Mathlib/Algebra/Lie/Derivation/Killing.lean | 41 | 43 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | theorem ae_differentiableWithinAt_of_mem_pi
{ι : Type*} [Fintype ι] {f : E → ι → ℝ} {s : Set E}
(hf : LipschitzOnWith C f s) : ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x := by
have A : ∀ i : ι, LipschitzWith 1 (fun x : ι → ℝ ↦ x i) := fun i => LipschitzWith.eval i
have : ∀ i : ι, ∀ᵐ x ∂μ, x ∈ s → Diffe... | Mathlib/Analysis/Calculus/Rademacher.lean | 354 | 364 |
/-
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.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... | theorem card_le_of_surjective [Finite α] (f : α → β) (hf : Function.Surjective f) :
Nat.card β ≤ Nat.card α := by
classical
| Mathlib/Data/Finite/Card.lean | 93 | 95 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Adjoin.Polynomial
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Ideal.Quotient.Operati... |
protected theorem freeAlgebra (ι : Type*) [Finite ι] : FiniteType R (FreeAlgebra R ι) := by
cases nonempty_fintype ι
classical
| Mathlib/RingTheory/FiniteType.lean | 78 | 81 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | theorem cyclotomic.dvd_X_pow_sub_one (n : ℕ) (R : Type*) [Ring R] :
cyclotomic n R ∣ X ^ n - 1 := by
suffices cyclotomic n ℤ ∣ X ^ n - 1 by
simpa only [map_cyclotomic_int, Polynomial.map_sub, Polynomial.map_one, Polynomial.map_pow,
Polynomial.map_X] using map_dvd (Int.castRingHom R) this
rcases n.eq_z... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 336 | 345 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | measurableSet _ := Metric.isClosed_closedBall.measurableSet
ae_eventually_mem := by
filter_upwards with y
filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
end MetricSpace
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 139 | 144 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yaël Dillies
-/
import Mathlib.Order.Interval.Set.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Tactic.Contrapose
import Mathl... |
theorem pow_log_le_add_one (b : ℕ) : ∀ x, b ^ log b x ≤ x + 1
| Mathlib/Data/Nat/Log.lean | 163 | 164 |
/-
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.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | Submodule.span R ((x ^ ·) '' Set.Iio i) := by
induction i with
| zero => simp [(minpolyDiv_monic hx).leadingCoeff]
| succ i IH =>
rw [coeff_minpolyDiv, add_sub_assoc, pow_succ, ← sub_mul, Algebra.algebraMap_eq_smul_one]
refine add_mem ?_ ?_
· apply Submodule.smul_mem
apply Submodule.subset... | Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 135 | 154 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... |
nonrec
theorem _root_.ContinuousWithinAt.clog {f : α → ℂ} {s : Set α} {x : α}
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 235 | 237 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.Topology.Category.CompHausLike.Limits
import Mathlib.Topology.Category.Stonean.Basic
/-!
# Explicit l... | Mathlib/Topology/Category/Stonean/Limits.lean | 105 | 112 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | theorem mul_right_surjective₀ {a : G₀} (h : a ≠ 0) : Surjective fun g => g * a := fun g =>
| Mathlib/Algebra/GroupWithZero/Basic.lean | 393 | 393 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | (∏ i ∈ s, z i ^ w i) ^ (∑ i ∈ s, w i)⁻¹ ≤ (∑ i ∈ s, w i * z i) / (∑ i ∈ s, w i) := by
convert geom_mean_le_arith_mean_weighted s (fun i => (w i) / ∑ i ∈ s, w i) z ?_ ?_ hz using 2
· rw [← finset_prod_rpow _ _ (fun i hi => rpow_nonneg (hz _ hi) _) _]
refine Finset.prod_congr rfl (fun _ ih => ?_)
rw [di... | Mathlib/Analysis/MeanInequalities.lean | 152 | 162 |
/-
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.LinearAlgebra.Matrix.Gershgorin
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
import Mathlib.NumberTheory.NumberField.Units.Basic... | simp_rw [Pi.norm_def, NNReal.coe_le_coe, Finset.sup_le_iff, ← NNReal.coe_le_coe] at h
exact h w (mem_univ _)
open scoped Classical in
theorem log_le_of_logEmbedding_le {r : ℝ} {x : (𝓞 K)ˣ} (hr : 0 ≤ r)
(h : ‖logEmbedding K (Additive.ofMul x)‖ ≤ r) (w : InfinitePlace K) :
|Real.log (w x)| ≤ (Fintype.card (... | Mathlib/NumberTheory/NumberField/Units/DirichletTheorem.lean | 128 | 151 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | simp_rw [ContinuousWithinAt, hg.tendsto_nhds_iff]; rfl
@[deprecated (since := "2024-10-28")]
alias Inducing.continuousWithinAt_iff := IsInducing.continuousWithinAt_iff
| Mathlib/Topology/ContinuousOn.lean | 1,289 | 1,292 |
/-
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.SetTheory.Cardinal.ENat
/-!
# Projection from cardinal numbers to natural numbers
In this file we define `Cardinal.toNat` to be the natural projectio... | Mathlib/SetTheory/Cardinal/ToNat.lean | 134 | 134 | |
/-
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... | rw [← Category.id_comp f, h, zero_comp]
· intro h
rw [← IsSplitMono.id f]
simp only [h, zero_comp]
theorem iff_isSplitEpi_eq_zero {X Y : C} (f : X ⟶ Y) [IsSplitEpi f] : IsZero Y ↔ f = 0 := by
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 189 | 194 |
/-
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.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,034 | 1,035 | |
/-
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.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... | Mathlib/Algebra/Order/Monovary.lean | 63 | 63 | |
/-
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... | Mathlib/Data/Num/Lemmas.lean | 1,539 | 1,539 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | fun x => match x with
| Option.none => none
| Option.some n => some n
instance : Coe ℕ∞ PartENat := ⟨ofENat⟩
| Mathlib/Data/Nat/PartENat.lean | 561 | 565 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... |
theorem opow_limit {a b : Ordinal} (ha : a ≠ 0) (hb : IsLimit b) :
a ^ b = ⨆ x : Iio b, a ^ x.1 := by
| Mathlib/SetTheory/Ordinal/Exponential.lean | 58 | 60 |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kim Morrison
-/
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Ring.Real
/-!
# The unit ... |
protected theorem prod_mem {ι : Type*} {t : Finset ι} {f : ι → ℝ}
| Mathlib/Topology/UnitInterval.lean | 229 | 230 |
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.Linea... | have : IsRoot (minpoly K σ.toLinearMap) ζ := by
simpa [minpoly_algEquiv_toLinearMap σ (isOfFinOrder_of_finite σ), hσ',
sub_eq_zero, IsGalois.card_aut_eq_finrank] using hζ.pow_eq_one
obtain ⟨v, hv⟩ := (Module.End.hasEigenvalue_of_isRoot this).exists_hasEigenvector
have hv' := hv.pow_apply
simp_rw [← Al... | Mathlib/FieldTheory/KummerExtension.lean | 472 | 481 |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.... | IsNat a a' → IsNat b b' → Nat.sub a' b' = c → IsNat (a - b) c
| _, _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, rfl => ⟨by simp⟩
| Mathlib/Tactic/NormNum/Basic.lean | 515 | 517 |
/-
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.Analysis.Convex.Hull
/-!
# Extreme sets
This file defines extreme sets and extreme points for sets in a module.
An extreme s... |
section LinearOrderedRing
variable [Ring 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [AddCommGroup E] [Module 𝕜 E]
variable [DenselyOrdered 𝕜] [NoZeroSMulDivisors 𝕜 E] {A : Set E} {x : E}
/-- A useful restatement using `segment`: `x` is an extreme point iff the only (closed) segments
that contain it are those w... | Mathlib/Analysis/Convex/Extreme.lean | 240 | 247 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | rw [← natCast_zmod_val ((u * u⁻¹ : Units (ZMod (n + 1))) : ZMod (n + 1))]
rw [Units.val_mul, val_mul, natCast_mod]
lemma isUnit_iff_coprime (m n : ℕ) : IsUnit (m : ZMod n) ↔ m.Coprime n := by
refine ⟨fun H ↦ ?_, fun H ↦ (unitOfCoprime m H).isUnit⟩
| Mathlib/Data/ZMod/Basic.lean | 782 | 786 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Order.... | rw [geom_sum_of_lt_one h1, div_lt_iff₀, inv_mul_cancel₀, tsub_lt_self_iff]
· exact ⟨h0.trans h1, pow_pos h0 n⟩
· rwa [ne_eq, tsub_eq_zero_iff_le, not_le]
· rwa [tsub_pos_iff_lt]
protected theorem Commute.mul_geom_sum₂_Ico [Ring R] {x y : R} (h : Commute x y) {m n : ℕ}
(hmn : m ≤ n) :
((x - y) * ∑ i ∈ F... | Mathlib/Algebra/GeomSum.lean | 303 | 314 |
/-
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.Torsion
import Mathlib.Algebra.Polynomial.Smeval
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Data.NNRat.Order
import Mathlib.Gro... | section Neg
namespace Ring
open Polynomial
variable {R : Type*} [NonAssocRing R] [Pow R ℕ] [BinomialRing R]
| Mathlib/RingTheory/Binomial.lean | 265 | 272 |
/-
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.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... |
@[fun_prop]
theorem hasFDerivAt_multiset_prod [DecidableEq ι] [Fintype ι] {u : Multiset ι} {x : ι → 𝔸'} :
HasFDerivAt (𝕜 := 𝕜) (fun x ↦ (u.map x).prod)
(Multiset.sum (u.map (fun i ↦ ((u.erase i).map x).prod • proj i))) x :=
hasStrictFDerivAt_multiset_prod.hasFDerivAt
| Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 641 | 646 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... |
@[simp]
theorem num_one : num (1 : RatFunc K) = 1 := by convert num_div (1 : K[X]) 1 <;> simp
@[simp]
theorem num_algebraMap (p : K[X]) : num (algebraMap _ _ p) = p := by convert num_div p 1 <;> simp
theorem num_div_dvd (p : K[X]) {q : K[X]} (hq : q ≠ 0) :
num (algebraMap _ _ p / algebraMap _ _ q) ∣ p := by
| Mathlib/FieldTheory/RatFunc/Basic.lean | 808 | 816 |
/-
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.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,081 | 1,084 | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Ring.NonZeroDivisors
import Mathlib.RingTheory.Localization.FractionRing
/-!
# The field of rational... | Mathlib/FieldTheory/RatFunc/Defs.lean | 228 | 232 | |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.FiberedCategory.HomLift
import Mathlib.CategoryTheory.Bicategory.Strict
import Mathlib.CategoryTheory.Functor.Category
import Ma... | lemma isHomLift {a : 𝒳.obj} {S : 𝒮} (ha : 𝒳.p.obj a = S) :
IsHomLift 𝒴.p (𝟙 S) (α.toNatTrans.app a) := by
subst ha; infer_instance
| Mathlib/CategoryTheory/FiberedCategory/BasedCategory.lean | 157 | 159 |
/-
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... |
open jacobiSym
/-- If `J(a | b)` is `-1`, then `a` is not a square modulo `b`. -/
theorem nonsquare_of_jacobiSym_eq_neg_one {a : ℤ} {b : ℕ} (h : J(a | b) = -1) :
¬IsSquare (a : ZMod b) := fun ⟨r, ha⟩ => by
rw [← r.coe_valMinAbs, ← Int.cast_mul, intCast_eq_intCast_iff', ← sq] at ha
apply (by norm_num : ¬(0 : ℤ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 266 | 276 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | has_compl {a} := m.measurableSet_compl a
has_iUnion_nat {f} _ hf := m.measurableSet_iUnion f hf
| Mathlib/MeasureTheory/PiSystem.lean | 564 | 565 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 457 | 469 | |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 314 | 314 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sébastien Gouëzel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
import Mathlib.Topology.FiberBundl... | def toFiberBundleCore : FiberBundleCore ι B F :=
{ Z with
coordChange := fun i j b => Z.coordChange i j b
continuousOn_coordChange := fun i j =>
isBoundedBilinearMap_apply.continuous.comp_continuousOn
((Z.continuousOn_coordChange i j).prodMap continuousOn_id) }
| Mathlib/Topology/VectorBundle/Basic.lean | 505 | 511 |
/-
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.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | use q, he.symm
obtain rfl | hq := eq_or_ne q 0
· rw [mul_zero] at he
subst he
simp
constructor
· conv_rhs => rw [he]
rw [(monic_multisetProd_X_sub_C p.roots).natDegree_mul' hq,
natDegree_multiset_prod_X_sub_C_eq_card]
· replace he := congr_arg roots he.symm
| Mathlib/Algebra/Polynomial/Roots.lean | 670 | 679 |
/-
Copyright (c) 2024 Colva Roney-Dougal. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Colva Roney-Dougal, Inna Capdeboscq, Susanna Fishel, Kim Morrison
-/
import Mathlib.GroupTheory.Nilpotent
import Mathlib.Order.Radical
/-!
# The Frattini subgroup
We give the def... | intro p p_prime P
-- The Frattini argument shows that the normalizer of `P` in `G`
-- together with `frattini G` generates `G`.
have frattini_argument := Sylow.normalizer_sup_eq_top P
-- and hence by the nongenerating property of the Frattini subgroup that
-- the normalizer of `P` in `G` is `G`.
have norm... | Mathlib/GroupTheory/Frattini.lean | 59 | 74 |
/-
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, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | rw [finrank, finrank, rank_finsupp, ← mk_toNat_eq_card, toNat_mul, toNat_lift, toNat_lift]
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 251 | 252 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | exact ⟨H₁, H₂⟩
have re_μ : ↑(RCLike.re μ) = μ := by
rw [← RCLike.conj_eq_iff_re]
exact hT.conj_eigenvalue_eq_self (hasEigenvalue_of_hasEigenvector key)
simpa [re_μ] using key
theorem hasEigenvalue_eigenvalues (i : Fin n) : HasEigenvalue T (hT.eigenvalues hn i) :=
Module.End.hasEigenvalue_of_hasEigenv... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 218 | 237 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | Mathlib/Algebra/CubicDiscriminant.lean | 594 | 598 | |
/-
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.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | (hfπ : ∀ i, f i ∈ π i) (hf : ∀ i, MeasurableSet (f i)) (hπ : iIndepSets π κ μ) :
iIndepSet f κ μ :=
(iIndepSet_iff_meas_biInter hf).2 fun _t ↦ hπ.meas_biInter _ fun _i _ ↦ hfπ _
variable {s t : Set Ω} (S T : Set (Set Ω))
theorem indepSet_iff_indepSets_singleton {m0 : MeasurableSpace Ω} (hs_meas : Measurable... | Mathlib/Probability/Independence/Kernel.lean | 803 | 810 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
import Mathlib.RingTheory.RingHom.FiniteType
import Mathlib.RingTheory.Spectrum.Prime.Jacobson
/-!
# Morphisms... | eq_affineLocally' := by
ext X Y f
rw [locallyOfFiniteType_iff, affineLocally_iff_affineOpens_le]
| Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean | 43 | 46 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | theorem num_denom_mul (x y : RatFunc K) :
(x * y).num * (x.denom * y.denom) = x.num * y.num * (x * y).denom :=
(num_mul_eq_mul_denom_iff (mul_ne_zero (denom_ne_zero x) (denom_ne_zero y))).mpr <| by
conv_lhs =>
rw [← num_div_denom x, ← num_div_denom y, div_mul_div_comm, ← RingHom.map_mul, ←
RingH... | Mathlib/FieldTheory/RatFunc/Basic.lean | 920 | 925 |
/-
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 | 1,830 | 1,833 | |
/-
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.Algebra.Order.Floor.Div
import Mathlib.Data.Nat.Factorization.Defs
/-!
# Roots of natural numbers, rounded up and down
This file defines the flooring and... | @[simp] lemma ceilRoot_zero_left (a : ℕ) : ceilRoot 0 a = 0 := by simp [ceilRoot]
| Mathlib/Data/Nat/Factorization/Root.lean | 120 | 120 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Limits.FunctorCategory.Finite
import M... | simp only [coimage_image_factorisation, PreservesKernel.iso_hom, Category.assoc,
kernel.lift_ι, Category.comp_id, PreservesCokernel.iso_inv,
cokernel.π_desc_assoc, Category.id_comp]
erw [kernelComparison_comp_ι _ ((evaluation C D).obj X)]
erw [π_comp_cokernelComparison_assoc _ ((evaluation C D).obj X)]
... | Mathlib/CategoryTheory/Abelian/FunctorCategory.lean | 64 | 76 |
/-
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.Algebra.Pi
import Mathlib.Algebra.Algebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
impo... |
/-- The automorphism of the polynomial algebra given by `p(X) ↦ p(a * X + b)`,
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 291 | 292 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... |
theorem Indep.isBasis_of_maximal_subset (hI : M.Indep I) (hIX : I ⊆ X)
(hmax : ∀ ⦃J⦄, M.Indep J → I ⊆ J → J ⊆ X → J ⊆ I) (hX : X ⊆ M.E := by aesop_mat) :
M.IsBasis I X := by
rw [isBasis_iff (by aesop_mat : X ⊆ M.E), and_iff_right hI, and_iff_right hIX]
exact fun J hJ hIJ hJX ↦ hIJ.antisymm (hmax hJ hIJ hJX... | Mathlib/Data/Matroid/Basic.lean | 897 | 902 |
/-
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.Module.NatInt
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.FreeGroup.Basic
/-!
# Free abelian groups
The free abelian group on ... | instance mul : Mul (FreeAbelianGroup α) :=
⟨fun x ↦ lift fun x₂ ↦ lift (fun x₁ ↦ of (x₁ * x₂)) x⟩
variable {α}
| Mathlib/GroupTheory/FreeAbelianGroup.lean | 387 | 390 |
/-
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... |
The usual Roth number corresponds to `addRothNumber (Finset.range n)`, see `rothNumberNat`."]
def mulRothNumber : Finset α →o ℕ :=
⟨fun s ↦ Nat.findGreatest (fun m ↦ ∃ t ⊆ s, #t = m ∧ ThreeGPFree (t : Set α)) #s, by
rintro t u htu
refine Nat.findGreatest_mono (fun m => ?_) (card_le_card htu)
| Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 256 | 261 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Sites.CoverPreserving
/-! Localization
In this file, given a Grothendieck topology `J` on a cat... | change _ ∈ J _ at h₁ ⊢
rw [Sieve.overEquiv_pullback]
exact J.pullback_stable _ h₁
| Mathlib/CategoryTheory/Sites/Over.lean | 127 | 129 |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
nonrec lemma StrongConcaveOn.strictConcaveOn (hf : StrongConcaveOn s m f) (hm : 0 < m) :
| Mathlib/Analysis/Convex/Strong.lean | 142 | 143 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | theorem prod_eq : f ×ˢ g = (f.map Prod.mk).seq g := f.map_prod id g
theorem prod_inf_prod {f₁ f₂ : Filter α} {g₁ g₂ : Filter β} :
(f₁ ×ˢ g₁) ⊓ (f₂ ×ˢ g₂) = (f₁ ⊓ f₂) ×ˢ (g₁ ⊓ g₂) := by
simp only [prod_eq_inf, comap_inf, inf_comm, inf_assoc, inf_left_comm]
theorem inf_prod {f₁ f₂ : Filter α} : (f₁ ⊓ f₂) ×ˢ g = (... | Mathlib/Order/Filter/Prod.lean | 356 | 364 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... |
section CharTwo
variable [CharP R 2]
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 390 | 393 |
/-
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... | Mathlib/MeasureTheory/Group/Measure.lean | 957 | 971 | |
/-
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 | 3,086 | 3,087 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Group.Submonoid.Basic
import M... | map (uncurry (· * ·)) (𝓝 (x₀, y₀)) = map (uncurry (· * ·)) (𝓝 x₀ ×ˢ 𝓝 y₀) := by
rw [nhds_prod_eq]
_ = map (fun p : M × M => x₀ * p.1 * (p.2 * y₀)) (𝓝 1 ×ˢ 𝓝 1) := by
-- Porting note: `rw` was able to prove this
| Mathlib/Topology/Algebra/Monoid.lean | 217 | 220 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.GaloisConnection.Defs
/-!
# Modular Lattices
This file defines (... | end UpperModular
section LowerModular
| Mathlib/Order/ModularLattice.lean | 151 | 153 |
/-
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... | theorem coe_add : ((φ + ψ : MvPolynomial σ R) : MvPowerSeries σ R) = φ + ψ :=
rfl
@[simp, norm_cast]
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 818 | 821 |
/-
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, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.DiscreteValuationRing.Basi... | -(constantCoeff k φ)⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff k x.1 φ * coeff k x.2 φ⁻¹ else 0 := by
rw [inv_eq_inv_aux, coeff_inv_aux n (constantCoeff k φ)⁻¹ φ]
@[simp]
theorem constantCoeff_inv (φ : k⟦X⟧) : constantCoeff k φ⁻¹ = (constantCoeff k φ)⁻¹ :=
MvPowerSeries.constant... | Mathlib/RingTheory/PowerSeries/Inverse.lean | 140 | 147 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | | zero => simp
| of j x =>
simp_rw [DirectSum.coeAddMonoidHom_of, DirectSum.of]
-- Porting note: was in the previous `simp_rw`, no longer works
-- This used to be `rw`, but we need `erw` after https://github.com/leanprover/lean4/pull/2644
erw [DFinsupp.singleAddHom_apply]
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,270 | 1,275 |
/-
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
/-!
# ... | Mathlib/SetTheory/Ordinal/Basic.lean | 1,441 | 1,441 | |
/-
Copyright (c) 2023 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Matrix.Notation... | {ω : FractionalOperation D 2} (symmega : ω.IsSymmetric)
{r : Fin 2 → D} (rin : r ∈ (ω.tt ![![a, b], ![b, a]])) :
f ![a, b] < f r := by
rw [FractionalOperation.tt, Multiset.mem_map] at rin
rw [show r = ![r 0, r 1] by simp [← List.ofFn_inj]]
apply lt_of_le_of_ne (mcf.right (r 0) (r 1)).left
intro equ
... | Mathlib/Combinatorics/Optimization/ValuedCSP.lean | 130 | 149 |
/-
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.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
/-!
# The Karoubi envelope ... | subst h_X
simpa only [mk.injEq, heq_eq_eq, true_and, eqToHom_refl, comp_id, id_comp] using h_p
/-- A morphism `P ⟶ Q` in the category `Karoubi C` is a morphism in the underlying category
`C` which satisfies a relation, which in the preadditive case, expresses that it induces a
map between the corresponding "formal... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 60 | 66 |
/-
Copyright (c) 2020 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.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | ((measurableSet_of_differentiableAt_with_param _ hf).compl.inter (MeasurableSet.const _))
theorem measurable_fderiv_apply_const_with_param [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f.uncurry) (y : E) :
Measurable (fun (p : α × E) ↦ fderiv 𝕜 (f p.1) p.2 y) :=
(ContinuousLinearMap.measurable_a... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 912 | 918 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | /-- `Matrix.blockDiagonal'` as a `RingHom`. -/
@[simps]
def blockDiagonal'RingHom [∀ i, DecidableEq (m' i)] [Fintype o] [∀ i, Fintype (m' i)]
[NonAssocSemiring α] : (∀ i, Matrix (m' i) (m' i) α) →+* Matrix (Σ i, m' i) (Σ i, m' i) α :=
{ blockDiagonal'AddMonoidHom m' m' α with
| Mathlib/Data/Matrix/Block.lean | 685 | 689 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | /-- A twisted version of the relation `Λ * ↑ζ = log` in terms of complex sequences. -/
lemma convolution_twist_vonMangoldt {N : ℕ} (χ : DirichletCharacter ℂ N) :
(↗χ * ↗Λ) ⍟ ↗χ = ↗χ * ↗Complex.log := by
rw [← convolution_vonMangoldt_const_one, ← χ.mul_convolution_distrib, mul_one]
| Mathlib/NumberTheory/LSeries/Dirichlet.lean | 354 | 357 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | rw [descPochhammer_succ_left, ih, mul_comp, ← mul_assoc, ← descPochhammer_succ_left, sub_comp,
X_comp, natCast_comp]
rw [Nat.cast_add, Nat.cast_one, sub_add_eq_sub_sub_swap]
@[simp]
theorem descPochhammer_natDegree (n : ℕ) [NoZeroDivisors R] [Nontrivial R] :
(descPochhammer R n).natDegree = n :... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 301 | 312 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.PSeriesComplex
/-!
# Definition of the Riemann zeta function
## Main definitions:
* `rieman... | differentiableAt_hurwitzZetaEven _ hs'
/-- We have `ζ(0) = -1 / 2`. -/
| Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 134 | 136 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 500 | 503 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | cases x with
| top => exact (h'x rfl).elim
| coe x =>
have : x ≠ 0 := fun h => by simp [h] at hx
simp [← coe_rpow_of_ne_zero this, NNReal.rpow_add this]
theorem rpow_add_of_nonneg {x : ℝ≥0∞} (y z : ℝ) (hy : 0 ≤ y) (hz : 0 ≤ z) :
x ^ (y + z) = x ^ y * x ^ z := by
induction x using recTopCoe
· rcas... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 571 | 583 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | IsVonNBounded 𝕜 S ↔ ∀ x : ι → E, (∀ n, x n ∈ S) → Tendsto (ε • x) l (𝓝 0) :=
⟨fun hS _ hxS => hS.smul_tendsto_zero (Eventually.of_forall hxS) (le_trans hε nhdsWithin_le_nhds),
isVonNBounded_of_smul_tendsto_zero (by exact hε self_mem_nhdsWithin)⟩
end sequence
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 243 | 247 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.RingTheory.Coprime.Lemmas
/-!
# Charac... | (hR : ringChar R ≠ 0) : IsUnit (p : R) ↔ ¬p ∣ ringChar R := by
have hch := CharP.cast_eq_zero R (ringChar R)
have hp : p.Prime := Fact.out
constructor
· rintro h₁ ⟨q, hq⟩
rcases IsUnit.exists_left_inv h₁ with ⟨a, ha⟩
have h₃ : ¬ringChar R ∣ q := by
rintro ⟨r, hr⟩
rw [hr, ← mul_assoc, mul... | Mathlib/Algebra/CharP/CharAndCard.lean | 24 | 47 |
/-
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... | rw [← h, rpow_add' hx]; rwa [h]
theorem rpow_add_of_nonneg (hx : 0 ≤ x) (hy : 0 ≤ y) (hz : 0 ≤ z) :
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 194 | 196 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | rw [← mul_self_inj_of_nonneg (norm_nonneg I) zero_le_one, one_mul, ← norm_mul,
| Mathlib/Analysis/RCLike/Basic.lean | 686 | 686 |
/-
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.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | · simp only [if_true, add_tsub_cancel_right, eq_self_iff_true]
· intro i _hi hink
rw [coeff_monomial]
by_cases hik : i < k
· simp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, zero_mul]
· push_neg at hik
rw [if_neg]
contrapose! hink
exact (tsub_eq_iff_eq_add_of_le hi... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 67 | 80 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
/-!
# Centers of rings
-/
assert_not_exists RelIso Finset Subsemigroup Field
vari... | left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul]
right_assoc _ _ := by rw [mul_neg, ha.right_assoc, mul_neg, mul_neg]
end Set
| Mathlib/Algebra/Ring/Center.lean | 81 | 86 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
/-!
# Finite maps over `Multiset`
-/
universe u v w
open List
... | theorem induction_on {C : Finmap β → Prop} (s : Finmap β) (H : ∀ a : AList β, C ⟦a⟧) : C s := by
rcases s with ⟨⟨a⟩, h⟩; exact H ⟨a, h⟩
| Mathlib/Data/Finmap.lean | 134 | 136 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Data.Fintype.Card
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.End
import Mathlib.Data.Finset.N... |
variable [DecidableEq α] [Fintype α] {f g : Perm α}
| Mathlib/GroupTheory/Perm/Support.lean | 270 | 271 |
/-
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.Algebra.Homology.HomotopyCategory.HomologicalFunctor
import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
import Mathlib.Algebra.Homology.HomotopyCateg... | lemma quasiIso_eq_subcategoryAcyclic_W :
quasiIso C (ComplexShape.up ℤ) = (subcategoryAcyclic C).W := by
ext K L f
exact ((homologyFunctor C (ComplexShape.up ℤ) 0).mem_homologicalKernel_W_iff f).symm
| Mathlib/Algebra/Homology/DerivedCategory/Basic.lean | 94 | 97 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Matrices as a normed space
In this file we provide the fo... | have :=
@nnnorm_inner_le_nnnorm α _ _ _ _ ((WithLp.equiv 2 <| _ → α).symm fun j => star (A i j))
((WithLp.equiv 2 <| _ → α).symm fun k => B k j)
| Mathlib/Analysis/Matrix.lean | 612 | 614 |
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