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/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.Analysis.Normed.Module.Basic
/-! # A lemma about measurability with density under scalar multip... | (hf : Measurable f) {g : α → E} :
AEMeasurable g (μ.withDensity fun x => (f x : ℝ≥0∞)) ↔
AEMeasurable (fun x => (f x : ℝ) • g x) μ := by
constructor
· rintro ⟨g', g'meas, hg'⟩
have A : MeasurableSet { x : α | f x ≠ 0 } := (hf (measurableSet_singleton 0)).compl
refine ⟨fun x => (f x : ℝ) • g' x... | Mathlib/MeasureTheory/Integral/LebesgueNormedSpace.lean | 20 | 47 |
/-
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.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | theorem ordConnectedSection_subset : ordConnectedSection s ⊆ s :=
range_subset_iff.2 fun _ => ordConnectedComponent_subset <| Nonempty.some_mem _
theorem eq_of_mem_ordConnectedSection_of_uIcc_subset (hx : x ∈ ordConnectedSection s)
(hy : y ∈ ordConnectedSection s) (h : [[x, y]] ⊆ s) : x = y := by
rcases hx wit... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 127 | 133 |
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.ZPowers.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.GroupAction.Defs
import Mathlib.GroupTheory.Subgroup.Centr... | Mathlib/GroupTheory/GroupAction/ConjAct.lean | 354 | 358 | |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | @[simp]
theorem range_face_points {n : ℕ} (s : Simplex k P n) {fs : Finset (Fin (n + 1))} {m : ℕ}
(h : #fs = m + 1) : Set.range (s.face h).points = s.points '' ↑fs := by
rw [face_points', Set.range_comp, Finset.range_orderEmbOfFin]
| Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 835 | 839 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... | theorem riemannZeta_two : riemannZeta 2 = (π : ℂ) ^ 2 / 6 := by
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_two.tsum_eq
· rw [← Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp [push_cast]
· norm_cast
theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 := by
| Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 211 | 217 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | mul_self_inj (norm_nonneg _) (norm_nonneg _), ← real_inner_self_eq_norm_mul_norm, ←
real_inner_self_eq_norm_mul_norm] at hd
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 571 | 572 |
/-
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... | show val (Nat.unaryCast n) = _ by induction n <;> simp [Nat.unaryCast, *]
| Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 434 | 435 |
/-
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.Data.Real.Basic
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Algebra.Order.AbsoluteValue.Euclidean
/-!
# Admissible absolute values
This file defi... | refine ⟨0, 1, ?_, ?_⟩
· simp
rintro ⟨i, ⟨⟩⟩
| succ n ih =>
intro ε hε b hb A
let M := h.card ε
-- By the "nicer" pigeonhole principle, we can find a collection `s`
-- of more than `M^n` remainders where the first components lie close together:
obtain ⟨s, s_inj, hs⟩ :
∃ s : Fin (M ^ n).succ →... | Mathlib/NumberTheory/ClassNumber/AdmissibleAbsoluteValue.lean | 73 | 119 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,714 | 1,721 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | have : x ⊓ yᶜ = ⊥ := of_not_not fun hbot =>
have ⟨a, ha, hle⟩ := (eq_bot_or_exists_atom_le _).resolve_left hbot
have ⟨hx, hy'⟩ := le_inf_iff.1 hle
have hy := h a ha hx
| Mathlib/Order/Atoms.lean | 471 | 474 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... |
@[simp]
| Mathlib/Algebra/TrivSqZeroExt.lean | 793 | 794 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | Mathlib/Algebra/Lie/Submodule.lean | 1,389 | 1,391 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,307 | 1,311 | |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | rw [← (CategoryTheory.shiftFunctor C (1 : ℤ)).map_eq_zero_iff, h]
lemma mor₃_eq_zero_of_epi₂ (h : Epi T.mor₂) : T.mor₃ = 0 := (T.mor₃_eq_zero_iff_epi₂ hT).2 h
lemma mor₂_eq_zero_of_epi₁ (h : Epi T.mor₁) : T.mor₂ = 0 := (T.mor₂_eq_zero_iff_epi₁ hT).2 h
| Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 279 | 282 |
/-
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.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Me... |
/-- The parameterized integral `x ↦ ∫ y, g (y⁻¹ * x) ∂μ` depends continuously on `y` when `g` is a
compactly supported continuous function on a topological group `G`, and `μ` is finite on compact
sets. -/
@[to_additive]
lemma continuous_integral_apply_inv_mul
{G : Type*} [TopologicalSpace G] [LocallyCompactSpace G... | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | 98 | 122 |
/-
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... | is preserved under function symbols, then `p` holds for all elements of the closure of `s`. -/
@[elab_as_elim]
theorem closure_induction {p : M → Prop} {x} (h : x ∈ closure L s) (Hs : ∀ x ∈ s, p x)
(Hfun : ∀ {n : ℕ} (f : L.Functions n), ClosedUnder f (setOf p)) : p x :=
(@closure_le L M _ ⟨setOf p, fun {_} => Hfu... | Mathlib/ModelTheory/Substructures.lean | 325 | 330 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Basic
/-!
# Preserving binary products
Constructions to relate... | /-- If the product comparison maps of `G` at every pair `(X,Y)` is an
isomorphism, then `G` preserves binary products. -/
lemma preservesBinaryProducts_of_isIso_prodComparison
[HasBinaryProducts C] [HasBinaryProducts D]
| Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean | 107 | 110 |
/-
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... | Mathlib/MeasureTheory/Function/L2Space.lean | 316 | 325 | |
/-
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... | · exact h' x hx
| Mathlib/GroupTheory/Perm/Support.lean | 301 | 301 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 287 | 302 | |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,460 | 2,462 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | dsimp at hk
rw [← hk]
simp [sizeUpTo_succ', k.is_lt]
| Mathlib/Combinatorics/Enumerative/Composition.lean | 357 | 359 |
/-
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... | Mathlib/Data/Nat/PartENat.lean | 851 | 858 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Alge... | simp only [eqToHom_trans]
@[reassoc (attr := simp)]
lemma XIsoOfEq_inv_comp_XIsoOfEq_hom (K : HomologicalComplex V c) {p₁ p₂ p₃ : ι}
(h₂₁ : p₂ = p₁) (h₂₃ : p₂ = p₃) :
(K.XIsoOfEq h₂₁).inv ≫ (K.XIsoOfEq h₂₃).hom = (K.XIsoOfEq (h₂₁.symm.trans h₂₃)).hom := by
| Mathlib/Algebra/Homology/HomologicalComplex.lean | 110 | 115 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | f '' {x | Minimal (· ∈ s) x} = {x | Maximal (· ∈ f '' s) x} :=
image_antitone_setOf_minimal hf
| Mathlib/Order/Minimal.lean | 497 | 498 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import... | exact (hab.trans hu.1.2).false
· push_neg at ha hb
replace hb : f b < f ⊥ := lt_of_le_of_ne hb <| hf_i.ne (lt_of_lt_of_le' hab bot_le).ne'
obtain ⟨u, hu⟩ := intermediate_value_Ioo' hab.le hf_c.continuousOn ⟨hb, ha⟩
| Mathlib/Topology/Order/IntermediateValue.lean | 612 | 615 |
/-
Copyright (c) 2021 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster, Eric Wieser
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.FreeAlgebra
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.SimpleRing.B... | /-- If a ring homomorphism `R →+* A` is injective and `R` has characteristic zero
then so does `A`. -/
theorem charZero_of_injective_ringHom {R A : Type*} [NonAssocSemiring R] [NonAssocSemiring A]
{f : R →+* A} (h : Function.Injective f) [CharZero R] : CharZero A where
| Mathlib/Algebra/CharP/Algebra.lean | 64 | 67 |
/-
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.Basic
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.Analysis.Calculus.FDeriv.Prod
/-!
# Divergence inte... | (Hd : ∀ x ∈ (Box.Icc I) \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x) (i : Fin (n + 1)) :
HasIntegral.{0, u, u} I GP (fun x => f' x (Pi.single i 1)) BoxAdditiveMap.volume
(integral.{0, u, u} (I.face i) GP (fun x => f (i.insertNth (I.upper i) x))
BoxAdditiveMap.volume -
integral.{0, u, u... | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | 149 | 252 |
/-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Ring.Pi
import Mathlib... | Mathlib/Algebra/Order/Kleene.lean | 163 | 163 | |
/-
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... | refine lt_irrefl #p.roots.toFinset ?_
calc
| Mathlib/Algebra/Polynomial/Roots.lean | 586 | 587 |
/-
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... | constructor
| Mathlib/FieldTheory/RatFunc/Basic.lean | 929 | 929 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | (hkb : k ∣ b) : k = 1 :=
dvd_one.mp (isUnit_iff_dvd_one.mp <| coprime_iff_isRelPrime.mp h_ab_coprime hka hkb)
| Mathlib/Data/Nat/GCD/Basic.lean | 227 | 228 |
/-
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.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... |
@[deprecated (since := "2024-12-07")] alias map.isMaximal := IsMaximal.map_bijective
@[deprecated (since := "2024-12-07")] alias comap.isMaximal := IsMaximal.comap_bijective
@[deprecated (since := "2024-12-07")] alias RingEquiv.bot_maximal_iff := isMaximal_iff_of_bijective
end Bijective
| Mathlib/RingTheory/Ideal/Maps.lean | 498 | 504 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 971 | 972 | |
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Tactic.Field... | /-- A helper sequence that can be used to prove the equality of the recursive and the explicit
definition using a telescoping sum argument. -/
private def gosperCatalan (n j : ℕ) : ℚ :=
Nat.centralBinom j * Nat.centralBinom (n - j) * (2 * j - n) / (2 * n * (n + 1))
| Mathlib/Combinatorics/Enumerative/Catalan.lean | 72 | 75 |
/-
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.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
/-! # Formal power series (in one va... | split_ifs with h
· rw [h, order_eq_top, LinearMap.map_zero]
· rw [order_eq]
constructor <;> intro i hi
· simp only [Nat.cast_inj] at hi
rwa [hi, coeff_monomial_same]
· simp only [Nat.cast_lt] at hi
rw [coeff_monomial, if_neg]
exact ne_of_lt hi
/-- The order of the monomial `a*X^n` i... | Mathlib/RingTheory/PowerSeries/Order.lean | 197 | 208 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | Mathlib/Data/Nat/Factorization/Basic.lean | 977 | 981 | |
/-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | (continuousOn_of_forall_continuousAt fun x hx => ?_)
· refine (continuousAt_cpow_const_of_re_pos (Or.inl ?_) hu).comp continuous_ofReal.continuousAt
rw [ofReal_re]; exact hx.1
· refine (continuousAt_cpow_const_of_re_pos (Or.inl ?_) hv).comp
(continuous_const.sub continuous_ofReal).continuo... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 155 | 209 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | rw [ListBlank.nth_map, ListBlank.nth_modifyNth, proj, PointedMap.mk_val]
by_cases h' : k' = k
· subst k'
| Mathlib/Computability/TuringMachine.lean | 547 | 549 |
/-
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.Notation.Pi
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE
import Mathlib.Data.Finset.Lattice.Fo... | theorem Pi.exists_forall_pos_add_lt [ExistsAddOfLE α] [Finite ι] {x y : ι → α}
(h : ∀ i, x i < y i) : ∃ ε, 0 < ε ∧ ∀ i, x i + ε < y i := by
cases nonempty_fintype ι
cases isEmpty_or_nonempty ι
· obtain ⟨a, ha⟩ := exists_ne (0 : α)
obtain ha | ha := ha.lt_or_lt <;> obtain ⟨b, hb, -⟩ := exists_pos_add_of_lt... | Mathlib/Algebra/Order/Field/Pi.lean | 21 | 31 |
/-
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.MorphismProperty.Composition
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Category.Quiv
/-!
# Const... | simp only [Functor.comp_map, hw] at hw'
refine Functor.congr_inv_of_congr_hom _ _ _ ?_ ?_ hw'
all_goals apply Functor.congr_obj h
variable (W) in
/-- The canonical bijection between objects in a category and its
localization with respect to a morphism_property `W` -/
| Mathlib/CategoryTheory/Localization/Construction.lean | 183 | 189 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | theorem continuous_stereoInvFun (hv : ‖v‖ = 1) : Continuous (stereoInvFun hv) :=
continuous_induced_rng.2
((contDiff_stereoInvFunAux (m := 0)).continuous.comp continuous_subtype_val)
open scoped InnerProductSpace in
attribute [-simp] AddSubgroupClass.coe_norm Submodule.coe_norm in
theorem stereo_left_inv (hv : ‖... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 194 | 205 |
/-
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.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
/-!
# Slope of a function
In this file we define the slope of a function `f : k... | simp [sub_ne_zero.2 (Ne.symm hab)]
rw [add_comm]
| Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 99 | 100 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Dynamics.BirkhoffSum.Basic
import Mathlib.Algebra.Module.Basic
/-!
# Birkhoff average
In this file we define `birkhoffAverage f g n x` to be
$$
\fr... | theorem birkhoffAverage_congr_ring' (S : Type*) [DivisionSemiring S] [Module S M] :
birkhoffAverage (α := α) (M := M) R = birkhoffAverage S := by
ext; apply birkhoffAverage_congr_ring
| Mathlib/Dynamics/BirkhoffSum/Average.lean | 68 | 70 |
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Nat.Cast.Field
import Mathlib.NumberTheory.ArithmeticFunction
/-!
# The von Mangoldt Function
In ... | rw [this, sum_sub_distrib, ← sum_mul, ← Int.cast_sum, ← coe_mul_zeta_apply, eq_comm, sub_eq_self,
moebius_mul_coe_zeta]
| Mathlib/NumberTheory/VonMangoldt.lean | 140 | 141 |
/-
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.Pi.Basic
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
import Mathli... | @[to_additive]
theorem image_mulSingle_Ioc (i : ι) (a b : α i) :
| Mathlib/Order/Interval/Set/Pi.lean | 176 | 177 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... |
@[simp]
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 618 | 619 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
# Basic properties of the manifold Fréchet ... |
theorem mdifferentiableAt_iff_source_of_mem_source
[IsManifold I 1 M] {x' : M} (hx' : x' ∈ (chartAt H x).source) :
MDifferentiableAt I I' f x' ↔
| Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 257 | 260 |
/-
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.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | For example if `x` and `y` are terms of type `α` then `-x + y + y + x + -y` is a
\"typical\" element of `FreeAddGroup α`. In particular if `α` is empty
then `FreeAddGroup α` is isomorphic to the trivial group, and if `α` has one term
then `FreeAddGroup α` is isomorphic to `ℤ`.
If `α` has two or more terms then `FreeAdd... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 402 | 421 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 344 | 346 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Iterate
impor... | theorem tendsto_mod_div_atTop_nhds_zero_nat {m : ℕ} (hm : 0 < m) :
Tendsto (fun n : ℕ => ((n % m : ℕ) : ℝ) / (n : ℝ)) atTop (𝓝 0) := by
have h0 : ∀ᶠ n : ℕ in atTop, 0 ≤ (fun n : ℕ => ((n % m : ℕ) : ℝ)) n := by aesop
exact tendsto_bdd_div_atTop_nhds_zero h0
(.of_forall (fun n ↦ cast_le.mpr (mod_lt n hm).le... | Mathlib/Analysis/SpecificLimits/Basic.lean | 97 | 113 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | rw [compl_compl]
have L' : Tendsto (fun r : ℝ => μ (sᶜ ∩ ({x} + r • t)) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 0) :=
tendsto_addHaar_inter_smul_zero_of_density_zero μ sᶜ x L t ht h''t
have L'' : Tendsto (fun r : ℝ => μ ({x} + r • t) / μ ({x} + r • t)) (𝓝[>] 0) (𝓝 1) := by
apply tendsto_const_nhds.congr' _
... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 785 | 834 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,605 | 1,611 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | Mathlib/Data/Option/NAry.lean | 219 | 223 | |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
/-!
# Homotopy between paths
In this file,... | simp [eval]
end
| Mathlib/Topology/Homotopy/Path.lean | 83 | 85 |
/-
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... | rw [coeff_mul, this, Finset.sum_insert, Finset.sum_singleton, coeff_zero_eq_constantCoeff,
mul_comm, add_comm]
norm_num
/-- First coefficient of the `n`-th power of a power series. -/
lemma coeff_one_pow (n : ℕ) (φ : R⟦X⟧) :
coeff R 1 (φ ^ n) = n * coeff R 1 φ * (constantCoeff R φ) ^ (n - 1) := by
rcases... | Mathlib/RingTheory/PowerSeries/Basic.lean | 663 | 674 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Or... |
section Simplex
section OrderedSemiring
variable (𝕜) (ι : Type*) [Semiring 𝕜] [PartialOrder 𝕜] [Fintype ι]
/-- The standard simplex in the space of functions `ι → 𝕜` is the set of vectors with non-negative
| Mathlib/Analysis/Convex/Basic.lean | 594 | 601 |
/-
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... | @[reassoc (attr := simp)]
lemma add'_comp (f₁ f₂ : L.obj X ⟶ L.obj Y) (g : L.obj Y ⟶ L.obj Z) :
add' W f₁ f₂ ≫ g = add' W (f₁ ≫ g) (f₂ ≫ g) := by
obtain ⟨α, h₁, h₂⟩ := exists_leftFraction₂ L W f₁ f₂
obtain ⟨β, hβ⟩ := exists_leftFraction L W g
obtain ⟨γ, hγ⟩ := (RightFraction.mk _ α.hs β.f).exists_leftFraction... | Mathlib/CategoryTheory/Localization/CalculusOfFractions/Preadditive.lean | 178 | 194 |
/-
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.Order.ConditionallyCompleteLattice.Group
import Mathlib.Topology.MetricSpace.Isometry
/-!
# Metric space gluing
Gluing two metric spaces along ... | cases p <;> cases q <;> first |rfl|simp [Sum.dist, glueDist, dist_comm, add_comm,
add_left_comm, add_assoc]
private theorem Sum.dist_comm (x y : X ⊕ Y) : Sum.dist x y = Sum.dist y x := by
cases x <;> cases y <;> simp [Sum.dist, _root_.dist_comm, add_comm, add_left_comm, add_assoc]
| Mathlib/Topology/MetricSpace/Gluing.lean | 221 | 225 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... |
private theorem ghostFun_pow (m : ℕ) : ghostFun (x ^ m) = ghostFun x ^ m := by
ghost_fun_tac X 0 ^ m, ![x.coeff]
| Mathlib/RingTheory/WittVector/Basic.lean | 196 | 199 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Defs
/-!
# Locus of unequal values of finitely supported dependent functions
Let `N : α → Type*` be a type family, assume that `N ... |
@[simp]
theorem neLocus_add_right [∀ a, AddRightCancelMonoid (N a)] (f g h : Π₀ a, N a) :
(f + h).neLocus (g + h) = f.neLocus g :=
zipWith_neLocus_eq_right _ _ _ _ fun _a ↦ add_left_injective
| Mathlib/Data/DFinsupp/NeLocus.lean | 110 | 114 |
/-
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 | 1,262 | 1,268 | |
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... |
This is one of the easy parts of the inverse function theorem: it assumes that we already have
an inverse function. -/
theorem HasFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
| Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 378 | 381 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,076 | 1,086 | |
/-
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... | end Semiring
section Ring
| Mathlib/RingTheory/FiniteType.lean | 533 | 535 |
/-
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.Order.Ring.Rat
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.PNat.Basic
import Mathlib.Data.PNat.Interval
import Mathlib.Tactic.NormNum... | theorem admissible_of_one_lt_sumInv {p q r : ℕ+} (H : 1 < sumInv {p, q, r}) :
Admissible {p, q, r} := by
simp only [Admissible]
let S := sort ((· ≤ ·) : ℕ+ → ℕ+ → Prop) {p, q, r}
have hS : S.Sorted (· ≤ ·) := sort_sorted _ _
have hpqr : ({p, q, r} : Multiset ℕ+) = S := (sort_eq LE.le {p, q, r}).symm
rw [h... | Mathlib/NumberTheory/ADEInequality.lean | 229 | 248 |
/-
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.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... |
@[simp]
theorem Iso.inv_hom (f : X ≅ Y) : inv f.hom = f.inv := by
apply inv_eq_of_hom_inv_id
| Mathlib/CategoryTheory/Iso.lean | 366 | 369 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mat... | intro _ ha
obtain ⟨i, hi, rfl⟩ := Multiset.mem_map.mp ha
exact hps i hi
theorem le_prod_of_submultiplicative_of_nonneg {M : Type*} [CommMonoid M]
(f : M → R) (h_nonneg : ∀ a, 0 ≤ f a) (h_one : f 1 ≤ 1)
(h_mul : ∀ x y : M, f (x * y) ≤ f x * f y) (s : Finset ι) (g : ι → M) :
f (∏ i ∈ s, g i) ≤ ∏ i ∈ s,... | Mathlib/Algebra/Order/BigOperators/Ring/Finset.lean | 125 | 141 |
/-
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.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | rw [← colimit.w _ e.op]
delta Cover.toMultiequalizer
rw [ConcreteCategory.comp_apply, ConcreteCategory.comp_apply]
apply congr_arg
dsimp [diagram]
apply Concrete.multiequalizer_ext (C := D)
intro i
simp only [← ConcreteCategory.comp_apply, Category.assoc, Multiequalizer.lift_ι, Category.comp_id,
Meq... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 161 | 173 |
/-
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... | lemma mul_convolution_distrib {R : Type*} [CommSemiring R] {n : ℕ} (χ : DirichletCharacter R n)
(f g : ℕ → R) :
| Mathlib/NumberTheory/LSeries/Dirichlet.lean | 120 | 121 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | have h' := fun x (hx : x ∈ [[a, b]]) => ne_of_mem_of_not_mem hx h
rw [integral_deriv_eq_sub' _ deriv_log' (fun x hx => differentiableAt_log (h' x hx))
(continuousOn_inv₀.mono <| subset_compl_singleton_iff.mpr h),
log_div (h' b right_mem_uIcc) (h' a left_mem_uIcc)]
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 441 | 444 |
/-
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.Algebra.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | simp? [SqLe, mul_assoc] at xy zw says simp only [SqLe, mul_assoc] at xy zw
simp [SqLe, mul_add, mul_comm, mul_left_comm, add_le_add, *]
theorem sqLe_cancel {c d x y z w : ℕ} (zw : SqLe y d x c) (h : SqLe (x + z) c (y + w) d) :
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 362 | 365 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 789 | 790 | |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... |
lemma hom_heq_of_cons_eq_cons {p : Path a b} {p' : Path a c}
| Mathlib/Combinatorics/Quiver/Path.lean | 50 | 51 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | · intro x hx; rw [← hf.zero_eq x hx, ← hg.zero_eq x hx]; rfl
· intro m hm x hx
convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x
((hf.fderivWithin m hm x hx).prodMk (hg.fderivWithin m hm x hx))
· intro m hm
exact (L m).continuous.comp_continuousOn ((hf.cont m hm).prodMk (hg.cont m hm))
@[depreca... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 481 | 488 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,585 | 1,585 | |
/-
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... | @[to_additive]
theorem IsOfFinOrder.of_mem_zpowers (h : IsOfFinOrder x) (h' : y ∈ Subgroup.zpowers x) :
IsOfFinOrder y := by
obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp h'
| Mathlib/GroupTheory/OrderOfElement.lean | 634 | 637 |
/-
Copyright (c) 2021 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.RingTheory.Jacobson.Ring
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.Spectrum.Prime.Basic
/-!
# Nullstellensatz
This file establish... | fun _ hp =>
(PrimeSpectrum.mem_vanishingIdeal _ _).2 fun _ hI =>
let ⟨x, hx⟩ := hI
hx.2 ▸ fun _ hx' => (Set.mem_singleton_iff.1 hx').symm ▸ hp x hx.1
theorem pointToPoint_zeroLocus_le (I : Ideal (MvPolynomial σ k)) :
pointToPoint '' MvPolynomial.zeroLocus I ≤ PrimeSpectrum.zeroLocus ↑I := fun J... | Mathlib/RingTheory/Nullstellensatz.lean | 131 | 140 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Semisimple.Defs
import Mathlib.LinearAlgebra.BilinearForm.Orthogonal
/-!
# Lie algebras with non-degenerate invariant bilinear forms are s... |
include hΦ_inv
lemma restrict_nondegenerate (I : LieIdeal K L) (hI : IsAtom I) :
(Φ.restrict I).Nondegenerate := by
rw [LinearMap.BilinForm.restrict_nondegenerate_iff_isCompl_orthogonal hΦ_refl]
| Mathlib/Algebra/Lie/InvariantForm.lean | 142 | 147 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | exact T.continuous.comp_aestronglyMeasurable (lpMeas.aestronglyMeasurable (condExpL2 E' 𝕜 hm f))
@[deprecated (since := "2025-01-21")]
alias condexpL2_comp_continuousLinearMap := condExpL2_comp_continuousLinearMap
variable {𝕜 𝕜'}
section CondexpL2Indicator
variable (𝕜)
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 322 | 332 |
/-
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.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import ... | obtain ⟨s, hs, ha, hb⟩ := hG.2 h
exact ⟨s.map f, hs.map, mem_map_of_mem _ ha, mem_map_of_mem _ hb⟩
@[simp] lemma locallyLinear_comap {G : SimpleGraph β} {e : α ≃ β} :
(G.comap e).LocallyLinear ↔ G.LocallyLinear := by
| Mathlib/Combinatorics/SimpleGraph/Triangle/Basic.lean | 74 | 78 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | RCLike.inner_apply, conj_trivial, ofReal_sum, ofReal_mul, Finset.mul_sum, neg_mul,
Finset.sum_neg_distrib, mul_assoc, add_left_inj, neg_inj]
norm_cast
rw [sq_sqrt]
· simp [Finset.mul_sum, mul_comm]
· exact Finset.sum_nonneg (fun i _hi ↦ by positivity)
| Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 265 | 271 |
/-
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 | 1,973 | 1,975 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | @[to_additive]
theorem Subgroup.eq_bot_or_eq_top_of_prime_card
(H : Subgroup G) [hp : Fact (Nat.card G).Prime] : H = ⊥ ∨ H = ⊤ := by
have : Finite G := Nat.finite_of_card_ne_zero hp.1.ne_zero
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 158 | 161 |
/-
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.Tilted
/-!
# Log-likelihood Ratio
The likelihood ratio between two measures `μ` and `ν` is their Radon-Nikodym derivative
`μ.rnDeri... | have hμ : μ = 0 := by ext s _; exact hμν (by simp [h])
simp only [hμ, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot
| inr h0 =>
filter_upwards [hμν.ae_le (toReal_rnDeriv_tilted_right μ ν hf), Measure.rnDeriv_pos hμν,
hμν.ae_le (Measure.rnDeriv_lt_top μ ν)] with x hx hx_pos hx_lt_top
... | Mathlib/MeasureTheory/Measure/LogLikelihoodRatio.lean | 176 | 181 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... |
variable {R} in
lemma congr {q : ℕ} (h : p = q) : CharP R q := h ▸ ‹CharP R p›
@[simp] lemma cast_eq_zero : (p : R) = 0 := (cast_eq_zero_iff R p p).2 dvd_rfl
| Mathlib/Algebra/CharP/Defs.lean | 54 | 58 |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... |
variable [DivisionRing R] [StarRing R]
| Mathlib/Algebra/Star/SelfAdjoint.lean | 264 | 265 |
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
/-!
# Theory of monic polynomials
We give se... | @[simp]
theorem degree_le_zero_iff_eq_one (hp : p.Monic) : p.degree ≤ 0 ↔ p = 1 := by
rw [← hp.natDegree_eq_zero_iff_eq_one, natDegree_eq_zero_iff_degree_le_zero]
theorem natDegree_mul (hp : p.Monic) (hq : q.Monic) :
(p * q).natDegree = p.natDegree + q.natDegree := by
nontriviality R
apply natDegree_mul'
s... | Mathlib/Algebra/Polynomial/Monic.lean | 158 | 167 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Pi
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingT... | · rwa [degree_zero, eq_comm, degree_eq_bot, eq_comm] at h_deg_eq
· exact eq_of_degree_sub_lt_of_eval_index_eq s hvs
(lt_of_lt_of_le (degree_sub_lt h_deg_eq hf hlc) h_deg_le)
h_eval
end Indexed
end Polynomial
end PolynomialDetermination
| Mathlib/LinearAlgebra/Lagrange.lean | 119 | 128 |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Topology.Separation.Basic
/-!
# Pointwise convergence of indicator functions
In this file, we prove the equivalenc... |
lemma tendsto_ite {β : Type*} {p : ι → Prop} [DecidablePred p] {q : Prop} [Decidable q]
{a b : β} {F G : Filter β}
(haG : {a}ᶜ ∈ G) (hbF : {b}ᶜ ∈ F) (haF : principal {a} ≤ F) (hbG : principal {b} ≤ G) :
Tendsto (fun i ↦ if p i then a else b) L (if q then F else G) ↔ ∀ᶠ i in L, p i ↔ q := by
constructor <... | Mathlib/Topology/IndicatorConstPointwise.lean | 38 | 66 |
/-
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 | 1,967 | 2,001 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | ∃ n, Sum.inl b ∈ F a n ∧ ∀ m < n, k ≤ m → ∃ a₂, Sum.inr a₂ ∈ F a m by
rcases this _ h 0 (by simp [F]) with ⟨n, hn₁, hn₂⟩
exact ⟨_, ⟨⟨_, hn₁⟩, fun {m} mn => hn₂ m mn (Nat.zero_le _)⟩, hn₁⟩
intro a₁ h₁
apply @PFun.fixInduction _ _ _ _ _ _ h₁
intro a₂ h₂ IH k hk
rcases PFun.mem_fix_iff.... | Mathlib/Computability/Partrec.lean | 741 | 749 |
/-
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.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... |
lemma NNReal.ball_zero_eq_Ico (c : ℝ) :
Metric.ball (0 : ℝ≥0) c = Set.Ico 0 c.toNNReal := by
by_cases c_pos : 0 < c
| Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 110 | 113 |
/-
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.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 832 | 834 | |
/-
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... | apply Units.val_eq_one.1
simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
rw [IsRoot.def] at hroot
rw [← prod_cyclotomic_eq_X_pow_sub_one hpos (ZMod p), ← Nat.cons_self_properDivisors hpos.ne',
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 579 | 582 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... | Mathlib/Algebra/Ring/Subring/Basic.lean | 1,374 | 1,377 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | theorem det_one_sub_mul_comm (A : Matrix m n α) (B : Matrix n m α) :
det (1 - A * B) = det (1 - B * A) := by
rw [sub_eq_add_neg, ← Matrix.neg_mul, det_one_add_mul_comm, Matrix.mul_neg, ← sub_eq_add_neg]
/-- A special case of the **Matrix determinant lemma** for when `A = I`. -/
theorem det_one_add_replicateCol_m... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 406 | 413 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
@[fun_prop]
protected theorem ContinuousAt.piMap {Y : ι → Type*} [∀ i, TopologicalSpace (Y i)]
{f : ∀ i, π i → Y i} {x : ∀ i, π i} (hf : ∀ i, ContinuousAt (f i) (x i)) :
| Mathlib/Topology/Constructions.lean | 641 | 644 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... |
theorem Pi.compl_def [∀ i, HasCompl (π i)] (x : ∀ i, π i) :
xᶜ = fun i ↦ (x i)ᶜ :=
| Mathlib/Order/Basic.lean | 713 | 715 |
/-
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... | theorem intCast (n : ℤ) : mapFun f (n : 𝕎 R) = n :=
| Mathlib/RingTheory/WittVector/Basic.lean | 120 | 120 |
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