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 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | /-- The special-value formula `Γ(k + 1 + 1/2) = (2 * k + 1)‼ * √π / (2 ^ (k + 1))` for half-integer
values of the gamma function in terms of `Nat.doubleFactorial`. -/
lemma Real.Gamma_nat_add_one_add_half (k : ℕ) :
Gamma (k + 1 + 1 / 2) = (2 * k + 1 : ℕ)‼ * √π / (2 ^ (k + 1)) := by
induction k with
| zero => si... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 354 | 367 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | simp_rw [Set.mem_setOf_eq, convexBodySumFun, normAtPlace_apply]
rfl
theorem convexBodySum_neg_mem {x : mixedSpace K} (hx : x ∈ (convexBodySum K B)) :
-x ∈ (convexBodySum K B) := by
rw [Set.mem_setOf, convexBodySumFun_neg]
exact hx
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 337 | 343 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,248 | 2,249 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib... | · rw [tprod_eq_one_of_not_multipliable hg]
@[to_additive]
theorem tprod_le_one (h : ∀ i, f i ≤ 1) : ∏' i, f i ≤ 1 := by
by_cases hf : Multipliable f
| Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 193 | 197 |
/-
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... | rcases eq_zero_or_pos b with rfl | hb
· simp
| Mathlib/Data/Nat/GCD/Basic.lean | 58 | 59 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | Mathlib/LinearAlgebra/Trace.lean | 335 | 342 | |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | /-- Generalized eigenspaces for exponents at least `finrank K V` are equal to each other. -/
theorem genEigenspace_eq_genEigenspace_finrank_of_le [FiniteDimensional K V]
(f : End K V) (μ : K) {k : ℕ} (hk : finrank K V ≤ k) :
f.genEigenspace μ k = f.genEigenspace μ (finrank K V) :=
le_antisymm
(genEigenspa... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 343 | 365 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.A... | lemma abs_lt_of_sq_lt_sq (h : a ^ 2 < b ^ 2) (hb : 0 ≤ b) : |a| < b := by
| Mathlib/Algebra/Order/Ring/Abs.lean | 110 | 110 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,483 | 1,490 | |
/-
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.Properties
/-!
# Function field of integral schemes
We define the function field of an irreducible scheme as the stalk of the generic poi... | convert subset_closure_inter_of_isPreirreducible_of_isOpen _ H.base_open.isOpen_range _
· rw [Set.univ_inter, Set.image_univ]
· apply PreirreducibleSpace.isPreirreducible_univ (X := Y)
· exact ⟨_, trivial, Set.mem_range_self hX.2.some⟩
noncomputable instance stalkFunctionFieldAlgebra [IrreducibleSpace X] (x : ... | Mathlib/AlgebraicGeometry/FunctionField.lean | 83 | 93 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,727 | 1,728 | |
/-
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.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
/-!
# Cartan subalgebras
Cartan subalgebras are one of the most important concepts in Lie theor... | have : x ∈ H.normalizer := by simp [LieSubalgebra.mem_normalizer_iff, e]
exact hx (by rwa [LieSubalgebra.IsCartanSubalgebra.self_normalizing, e] at this)
instance (priority := 500) [Nontrivial L] (H : LieSubalgebra R L) [H.IsCartanSubalgebra] :
Nontrivial H := by
refine (subsingleton_or_nontrivial H).elim (f... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 97 | 102 |
/-
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.ShortComplex.Exact
import Mathlib.CategoryTheory.Shift.ShiftSequence
import Mathlib.CategoryTheory.Triangulated.Functor
import Mathlib.CategoryT... | rw [← F.shiftMap_comp, comp_distTriang_mor_zero₃₁ _ hT, shiftMap_zero]
@[reassoc]
lemma homologySequence_comp :
(F.shift n₀).map T.mor₁ ≫ (F.shift n₀).map T.mor₂ = 0 := by
rw [← Functor.map_comp, comp_distTriang_mor_zero₁₂ _ hT, Functor.map_zero]
| Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean | 197 | 202 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | revert s
induction n with | zero => _ | succ n IH => _ <;>
| Mathlib/Data/Seq/Computation.lean | 503 | 504 |
/-
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.Analysis.Complex.AbsMax
import Mathlib.Analysis.Asymptotics.SuperpolynomialDecay
/-!
# Phragmen-Lindelöf principle
In this file we prove several ve... | -- We are going to apply `PhragmenLindelof.horizontal_strip` to `f ∘ Complex.exp` and `ζ`.
change ‖(f ∘ exp) ζ‖ ≤ C
have H : MapsTo exp (im ⁻¹' Ioo 0 (π / 2)) (Ioi 0 ×ℂ Ioi 0) := fun z hz ↦ by
rw [mem_reProdIm, exp_re, exp_im, mem_Ioi, mem_Ioi]
have : 0 < Real.cos z.im := Real.cos_pos_of_mem_Ioo ⟨by linar... | Mathlib/Analysis/Complex/PhragmenLindelof.lean | 351 | 408 |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Math... |
protected theorem add_smul (y z : R[S⁻¹]) (x : X[S⁻¹]) :
(y + z) • x = y • x + z • x := by
induction' x with r₁ s₁
induction' y with r₂ s₂
induction' z with r₃ s₃
rcases oreDivAddChar' r₂ r₃ s₂ s₃ with ⟨ra, sa, ha, q⟩
rw [q]
clear q
rw [OreLocalization.expand' r₂ s₂ sa]
rcases oreDivSMulChar' (sa •... | Mathlib/RingTheory/OreLocalization/Ring.lean | 36 | 62 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | simp [roots_C_mul_X_add_C _ this]
theorem exists_root_of_degree_eq_one (h : degree p = 1) : ∃ x, IsRoot p x :=
⟨-((p.coeff 1)⁻¹ * p.coeff 0), by
rw [← mem_roots (by simp [← zero_le_degree_iff, h])]
simp [roots_degree_eq_one h]⟩
theorem coeff_inv_units (u : R[X]ˣ) (n : ℕ) : ((↑u : R[X]).coeff n)⁻¹ = (↑u⁻¹ ... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 502 | 509 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | rw [← Nat.cofinite_eq_atTop, Filter.eventually_cofinite, ← finite_image_iff this.injOn]
apply (finite_genWeightSpace_ne_bot R L M).subset
simp [f]
intro k l hkl
replace hkl : (k : ℤ) • χ₁ = (l : ℤ) • χ₁ := by
simpa only [f, add_left_inj, natCast_zsmul] using hkl
exact Nat.cast_inj.mp <| smul_left_... | Mathlib/Algebra/Lie/Weights/Chain.lean | 75 | 85 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | theorem ne_one_of_nnnorm_ne_zero {a : E} : ‖a‖₊ ≠ 0 → a ≠ 1 :=
mt <| by
rintro rfl
exact nnnorm_one'
| Mathlib/Analysis/Normed/Group/Basic.lean | 736 | 739 |
/-
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 ... | section Mul
| Mathlib/GroupTheory/FreeAbelianGroup.lean | 383 | 384 |
/-
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... | simp [descPochhammer_eval_zero, h]
theorem descPochhammer_succ_right (n : ℕ) :
descPochhammer R (n + 1) = descPochhammer R n * (X - (n : R[X])) := by
suffices h : descPochhammer ℤ (n + 1) = descPochhammer ℤ n * (X - (n : ℤ[X])) by
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 289 | 293 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Pointwise.Set.Sc... | refine Subset.antisymm ?_ (smul_preimage_set_subsetₛₗ f c t)
rw [subset_def, hc.forall]
rintro x ⟨y, hy, hxy⟩
rw [map_smulₛₗ, hc'.eq_iff] at hxy
subst y
exact smul_mem_smul_set hy
end SMul
section Monoid
variable {M N F : Type*} {α β : Type*} {σ : M → N} [Monoid M] [Monoid N]
[MulAction M α] [MulAction... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 72 | 84 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 492 | 496 | |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Damiano Testa, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Operations
import Mat... |
theorem degree_divX_lt (hp0 : p ≠ 0) : (divX p).degree < p.degree := by
haveI := Nontrivial.of_polynomial_ne hp0
calc
degree (divX p) < (divX p * X + C (p.coeff 0)).degree :=
if h : degree p ≤ 0 then by
have h' : C (p.coeff 0) ≠ 0 := by rwa [← eq_C_of_degree_le_zero h]
rw [eq_C_of_degree_... | Mathlib/Algebra/Polynomial/Inductions.lean | 103 | 111 |
/-
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.GeomSum
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.NoZeroSMulDiviso... | obtain ⟨n, hn⟩ := h
use n
rw [neg_pow, hn, mul_zero]
| Mathlib/RingTheory/Nilpotent/Basic.lean | 40 | 43 |
/-
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... |
end IntegrationByPartsBilinear
section IntegrationByPartsAlgebra
variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A]
{a : ℝ} {a' b' : A} {u : ℝ → A} {v : ℝ → A} {u' : ℝ → A} {v' : ℝ → A}
/-- For finite intervals, see: `intervalIntegral.integral_deriv_mul_eq_sub`. -/
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,223 | 1,231 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
@[simp]
theorem isCycleOn_one : (1 : Perm α).IsCycleOn s ↔ s.Subsingleton := by
simp [IsCycleOn, Set.bijOn_id, Set.Subsingleton]
alias ⟨IsCycleOn.subsingleton, _root_.Set.Subsingleton.isCycleOn_one⟩ := isCycleOn_one
@[simp]
theorem isCycleOn_singleton : f.IsCycleOn {a} ↔ f a = a := by simp [IsCycleOn, SameCycle.rf... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 680 | 701 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.Module.BigOperators
import Math... | {H F : Type*} [AddMonoid H] [FunLike F G H] [AddMonoidHomClass F G H] (f : F) :
A[G] →ₐ[k] A[H] :=
{ mapDomainRingHom A f with commutes' := mapDomain_algebraMap A f }
| Mathlib/Algebra/MonoidAlgebra/Basic.lean | 531 | 533 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | coe_pnat_injective.eq_iff
| Mathlib/Data/PNat/Prime.lean | 35 | 36 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | add_eq_left, natSepDegree_eq_zero_iff] at h
simpa only [eq_one_of_monic_natDegree_zero ((hM.pow _).of_mul_monic_left (hf ▸ hm)) h,
mul_one, ← hp] using hf
/-- A monic polynomial over a field `F` of exponential characteristic `q` has separable degree one
if and only if it is of the form `(X ^ (q ^ n) - C y) ^... | Mathlib/FieldTheory/SeparableDegree.lean | 602 | 613 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Bounded
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topology.MetricSpace.Thickening
/-!
# P... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 306 | 308 | |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 400 | 403 | |
/-
Copyright (c) 2022 Joachim Breitner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joachim Breitner
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.SupIndep
/-!
# Canonical homomorphism from a finite famil... | f.comp (MonoidHom.noncommPiCoprod ϕ hcomm) =
MonoidHom.noncommPiCoprod (fun i ↦ f.comp (ϕ i)) hcomm' :=
MonoidHom.ext fun _ ↦ by
simp only [MonoidHom.noncommPiCoprod, MonoidHom.coe_comp, MonoidHom.coe_mk, OneHom.coe_mk,
Function.comp_apply, Finset.map_noncommProd]
end MonoidHom
end FamilyOfMonoi... | Mathlib/GroupTheory/NoncommPiCoprod.lean | 193 | 204 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... | rintro (_ | _) <;> rfl
map_comp := by
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 281 | 282 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | ⟨fun h t hs => h t (mem_ssubsets.2 hs), fun h _ _ => h _ _⟩
/-- A version of `Finset.decidableExistsOfDecidableSSubsets` with a non-dependent `p`.
| Mathlib/Data/Finset/Powerset.lean | 158 | 160 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | Mathlib/Order/Irreducible.lean | 338 | 339 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
impo... | toRing := hf.ring f zero one add mul neg sub nsmul zsmul npow natCast intCast
__ := hf.groupWithZero f zero one mul inv div npow zpow
| Mathlib/Algebra/Field/Basic.lean | 236 | 237 |
/-
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.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.SuccPred.Limit
/-!
# Relation between `IsSuccPrelimit` and `iSup` in (conditionally) comple... | variable [ConditionallyCompleteLinearOrderBot α] {f : ι → α} {s : Set α} {x : α}
/-- See `csSup_mem_of_not_isSuccPrelimit` for the `ConditionallyCompleteLinearOrder` version. -/
lemma csSup_mem_of_not_isSuccPrelimit'
(hbdd : BddAbove s) (hlim : ¬ IsSuccPrelimit (sSup s)) : sSup s ∈ s := by
| Mathlib/Order/SuccPred/CompleteLinearOrder.lean | 82 | 86 |
/-
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... | DifferentiableAt 𝕜 (f ∘ iso) x ↔ DifferentiableAt 𝕜 f (iso x) := by
simp only [← differentiableWithinAt_univ, ← iso.comp_right_differentiableWithinAt_iff,
preimage_univ]
theorem comp_right_differentiableOn_iff {f : F → G} {s : Set F} :
DifferentiableOn 𝕜 (f ∘ iso) (iso ⁻¹' s) ↔ DifferentiableOn 𝕜 f s... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 172 | 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 only [HomologicalComplex.XIsoOfEq, eqToIso_refl, Iso.refl_hom, comp_id]
@[reassoc]
lemma v_comp_XIsoOfEq_inv
(γ : Cochain F G n) (p q q' : ℤ) (hpq : p + n = q) (hq' : q' = q) :
γ.v p q hpq ≫ (HomologicalComplex.XIsoOfEq G hq').inv = γ.v p q' (by rw [hq', hpq]) := by
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 211 | 216 |
/-
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.PartrecCode
import Mathlib.Data.Set.Subsingleton
/-!
# Computability theory and the halting problem
A universal partial recursive funct... | rw [← Option.some_inj, eq_comm]
rfl
| Mathlib/Computability/Halting.lean | 370 | 372 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Canonical
import Mathlib.Order.Interval.Basic
im... | Mathlib/Algebra/Order/Interval/Basic.lean | 710 | 715 | |
/-
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... | exact (isBoundedBilinearMap_apply.isBoundedLinearMap_left _).contDiff
· rintro ⟨h', h⟩
refine ⟨h', ?_⟩
have : fderivWithin 𝕜 f₂ s₂ = smulRight (1 : 𝕜 →L[𝕜] 𝕜) ∘ derivWithin f₂ s₂ := by
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,262 | 1,265 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | theorem polar_add (f g : M → N) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by
simp only [polar, Pi.add_apply]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 103 | 104 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | Mathlib/Data/Finset/Image.lean | 811 | 812 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | theorem log_neg (h0 : 0 < x) (h1 : x < 1) : log x < 0 :=
(log_neg_iff h0).2 h1
theorem log_neg_of_lt_zero (h0 : x < 0) (h1 : -1 < x) : log x < 0 := by
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 180 | 183 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | "A more general version of `finsum_mem_add_distrib` that only requires `s ∩ support f`
and `s ∩ support g` rather than `s` to be finite."]
theorem finprod_mem_mul_distrib' (hf : (s ∩ mulSupport f).Finite) (hg : (s ∩ mulSupport g).Finite) :
| Mathlib/Algebra/BigOperators/Finprod.lean | 554 | 556 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | theorem ContDiff.continuous_fderiv (h : ContDiff 𝕜 n f) (hn : 1 ≤ n) :
Continuous (fderiv 𝕜 f) :=
(contDiff_one_iff_fderiv.1 (h.of_le hn)).2
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,217 | 1,220 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | Mathlib/Data/Set/Lattice.lean | 2,177 | 2,181 | |
/-
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... |
/-- Continuity from below: the measure of the union of a sequence of (not necessarily measurable)
sets is the limit of the measures of the partial unions. -/
theorem tendsto_measure_iUnion_accumulate {α ι : Type*}
[Preorder ι] [IsCountablyGenerated (atTop : Filter ι)]
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 583 | 587 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | Mathlib/Analysis/Quaternion.lean | 244 | 247 | |
/-
Copyright (c) 2024 Etienne Marion. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Etienne Marion
-/
import Mathlib.Data.Finite.Prod
import Mathlib.MeasureTheory.SetSemiring
/-!
# Algebra of sets
In this file we define the notion of algebra of sets and give its bas... | /-- An algebra of sets is closed by finite intersections. -/
theorem biInter_mem {ι : Type*} (h𝒜 : IsSetAlgebra 𝒜) {s : ι → Set α} (S : Finset ι)
(hs : ∀ i ∈ S, s i ∈ 𝒜) : ⋂ i ∈ S, s i ∈ 𝒜 := by
by_cases h : S = ∅
· rw [h, ← Finset.set_biInter_coe, Finset.coe_empty, biInter_empty]
exact h𝒜.univ_mem
·... | Mathlib/MeasureTheory/SetAlgebra.lean | 86 | 92 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 1,818 | 1,821 | |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | ⨍ x in s, f x ∂μ = ∫ x, f x ∂(μ s)⁻¹ • μ.restrict s := by
simp only [average_eq', restrict_apply_univ]
| Mathlib/MeasureTheory/Integral/Average.lean | 336 | 337 |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.RingTheory.Regular.IsSMulRegular
import Mathlib.RingTheory.Artinian.Module
import Mathlib.RingTheory.Nakayama
import Mathlib.Algebra.Equiv.TransferInst... | Set.ext (fun _ => List.mem_map.symm)
lemma ofList_cons_smul {R} [CommSemiring R] (r : R) (rs : List R) {M}
[AddCommMonoid M] [Module R M] (N : Submodule R M) :
| Mathlib/RingTheory/Regular/RegularSequence.lean | 60 | 63 |
/-
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... | simp only [eval, eval₂_eq_sum, RingHom.id_apply]
have bound :=
calc
(p * (X - C r)).natDegree ≤ p.natDegree + (X - C r).natDegree := natDegree_mul_le
| Mathlib/Algebra/Polynomial/AlgebraMap.lean | 548 | 551 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,644 | 1,711 | |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | first one is not almost everywhere zero, then the second one is integrable. -/
theorem IndepFun.integrable_right_of_integrable_mul {β : Type*} [MeasurableSpace β] {X Y : Ω → β}
[NormedDivisionRing β] [BorelSpace β] (hXY : IndepFun X Y μ) (h'XY : Integrable (X * Y) μ)
(hX : AEStronglyMeasurable X μ) (hY : AEStro... | Mathlib/Probability/Integration.lean | 180 | 197 |
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNor... | let ⟨_, hpf⟩ := hf
hpf.const_smul.analyticWithinAt
@[fun_prop]
theorem AnalyticAt.fun_const_smul (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (fun z ↦ c • f z) x :=
let ⟨_, hpf⟩ := hf
hpf.const_smul.analyticAt
@[fun_prop]
theorem AnalyticAt.const_smul (hf : AnalyticAt 𝕜 f x) : AnalyticAt 𝕜 (c • f) x :=
hf.fun... | Mathlib/Analysis/Analytic/Constructions.lean | 212 | 236 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | (h_distrib : ∀ a b c, f (g a b) c = g' (f₁ a c) (f₂ b c)) :
image₂ f (image₂ g s t) u ⊆ image₂ g' (image₂ f₁ s u) (image₂ f₂ t u) :=
coe_subset.1 <| by
push_cast
exact Set.image2_distrib_subset_right h_distrib
| Mathlib/Data/Finset/NAry.lean | 386 | 391 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.PathCategory.Basic
/-!
# Free groupoid on a quiver
This file defines the free groupoid on a q... | Quot.mk (@Quotient.CompClosure _ _ redStep _ _) (𝟙 X) := by
apply Quot.eqvGen_sound
induction p with
| nil => apply EqvGen.refl
| cons q f ih =>
simp only [Quiver.Path.reverse]
fapply EqvGen.trans
-- Porting note: `Quiver.Path.*` and `Quiver.Hom.*` notation not working
· exact q ≫ Quiver.... | Mathlib/CategoryTheory/Groupoid/FreeGroupoid.lean | 81 | 90 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | exact convex_univ.eqOn_of_fderivWithin_eq hf.differentiableOn hg.differentiableOn
uniqueDiffOn_univ (fun x _ => by simpa using hf' _) (mem_univ _) hfgx
lemma isLittleO_pow_succ {x₀ : E} {n : ℕ} (hs : Convex ℝ s) (hx₀s : x₀ ∈ s)
(hff' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf' : f' =o[𝓝[s] x₀] fun x ↦ ‖... | Mathlib/Analysis/Calculus/MeanValue.lean | 645 | 649 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Products.Basic
/-!
# Curry and uncurry, as functors.
We define `curry : ((C × D) ⥤ E) ⥤ (C ⥤ (D ⥤ E)... | NatIso.ofComponents fun _ => Iso.refl _
variable (B C D E)
| Mathlib/CategoryTheory/Functor/Currying.lean | 120 | 123 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | rw [← Finset.coe_sup' hs, Finset.sup'_eq_sup]
| Mathlib/Algebra/MonoidAlgebra/Degree.lean | 263 | 263 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | theorem support_map : (map f p).support = f '' p.support :=
| Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 52 | 52 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | apply coeff_eq_zero_of_totalDegree_lt; rw [h]
exact Finset.sum_pos (fun i hi ↦ Nat.pos_of_ne_zero <| Finsupp.mem_support_iff.mp hi)
(Finsupp.support_nonempty_iff.mpr <| Ne.symm hm)
· rw [h, totalDegree_C]
theorem totalDegree_rename_le (f : σ → τ) (p : MvPolynomial σ R) :
(rename f p).totalDegree ≤ ... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 509 | 520 |
/-
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... | rw [coeff_mul, Finset.sum_eq_single_of_mem ⟨φ.order.toNat, ψ.order.toNat⟩ (by simp)]
· exact mul_ne_zero (coeff_order h.1) (coeff_order h.2)
· intro ij hij h
rcases trichotomy_of_add_eq_add (mem_antidiagonal.mp hij) with h' | h' | h'
| Mathlib/RingTheory/PowerSeries/Order.lean | 362 | 365 |
/-
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.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | forall₂_congr fun u _ => by simp
/-- A sequence of functions `Fₙ` converges uniformly to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ˢ ⊤` to the uniformity.
| Mathlib/Topology/UniformSpace/UniformConvergence.lean | 124 | 127 |
/-
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... | theorem addHaar_real_closedBall' (x : E) {r : ℝ} (hr : 0 ≤ r) :
μ.real (closedBall x r) = r ^ finrank ℝ E * μ.real (closedBall 0 1) := by
simp only [measureReal_def, addHaar_closedBall' μ x hr, ENNReal.toReal_mul, mul_eq_mul_right_iff,
| Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 474 | 476 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | have h : univ = insert circumcenterIndex (univ.map (pointIndexEmbedding n)) := by
ext x
refine ⟨fun h => ?_, fun _ => mem_univ _⟩
obtain i | - := x
· exact mem_insert_of_mem (mem_map_of_mem _ (mem_univ i))
· exact mem_insert_self _ _
change _ = (∑ i, f (pointIndexEmbedding n i)) + _
rw [add_co... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 435 | 445 |
/-
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.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | theorem lineDifferentiableWithinAt_univ :
LineDifferentiableWithinAt 𝕜 f univ x v ↔ LineDifferentiableAt 𝕜 f x v := by
simp only [LineDifferentiableWithinAt, LineDifferentiableAt, preimage_univ,
differentiableWithinAt_univ]
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 160 | 163 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.MonoidLocalization.Away
import Mathlib.Algebra.Algebra.Pi
import Mathlib.RingTheory.I... | end Away
end Away
variable [IsLocalization M S]
section AtUnits
variable (R) (S)
/-- The localization away from a unit is isomorphic to the ring. -/
noncomputable def atUnit (x : R) (e : IsUnit x) [IsLocalization.Away x S] : R ≃ₐ[R] S :=
atUnits R (Submonoid.powers x)
| Mathlib/RingTheory/Localization/Away/Basic.lean | 282 | 294 |
/-
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.Integral.IntegralEqImproper
/-!
# Integrals against peak functions
A sequence of peak functions is a sequence of functions with a... | apply tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux hs ht hts h'ts
hnφ hlφ hiφ h'iφ
· apply hmg.sub
simp only [integrable_indicator_iff ht, integrableOn_const, ht, Measure.restrict_apply]
right
exact lt_of_le_of_lt (measure_mono inter_subset_left) (h't.lt_top)
· rw [... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 192 | 225 |
/-
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... | Mathlib/CategoryTheory/Iso.lean | 591 | 593 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | variable [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) {x y : R}
theorem map_inv {R : Type*} [DivisionRing R] (v : Valuation R Γ₀) : ∀ x, v x⁻¹ = (v x)⁻¹ :=
map_inv₀ _
| Mathlib/RingTheory/Valuation/Basic.lean | 342 | 345 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... |
theorem ContDiffOn.continuousOn_fderivWithin (h : ContDiffOn 𝕜 n f s) (hs : UniqueDiffOn 𝕜 s)
(hn : 1 ≤ n) : ContinuousOn (fderivWithin 𝕜 f s) s :=
((contDiffOn_succ_iff_fderivWithin hs).1
(h.of_le (show 0 + (1 : WithTop ℕ∞) ≤ n from hn))).2.2.continuousOn
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 901 | 905 |
/-
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.GradedObject.Associator
import Mathlib.CategoryTheory.GradedObject.Single
/-!
# The left and right unitors
Given a bifunctor `F : C ⥤ D ⥤ D`, an ... | (mapBifunctorRightUnitorCofanIsColimit F Y e p hp X j)).symm)
@[reassoc (attr := simp)]
lemma ι_mapBifunctorRightUnitor_hom_apply (j : J) :
ιMapBifunctorMapObj F p X ((single₀ I).obj Y) j 0 j (hp j) ≫
(mapBifunctorRightUnitor F Y e p hp X).hom j =
(F.obj (X j)).map (singleObjApplyIso (0 : I) Y).h... | Mathlib/CategoryTheory/GradedObject/Unitor.lean | 217 | 224 |
/-
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... | theorem measure_union (hd : Disjoint s₁ s₂) (h : MeasurableSet s₂) : μ (s₁ ∪ s₂) = μ s₁ + μ s₂ :=
measure_union₀ h.nullMeasurableSet hd.aedisjoint
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 107 | 109 |
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
This file defines the harmonic numbers.
* `Mathilb/Num... | Mathlib/NumberTheory/Harmonic/Defs.lean | 37 | 41 | |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
/-!
# Destuttering of Lists
This file proves theorems about `List.destutter` (in `Data.List.Defs`), which greedily remov... |
@[simp]
theorem destutter_nil : ([] : List α).destutter R = [] :=
rfl
theorem destutter_cons' : (a :: l).destutter R = destutter' R a l :=
| Mathlib/Data/List/Destutter.lean | 109 | 114 |
/-
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.EngelSubalgebra
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Module.LinearMap.Polynomial
import Mathlib.LinearAlgebra.Ei... | lemma rank_eq_natTrailingDegree [Nontrivial R] [DecidableEq ι] :
rank R L M = (polyCharpoly φ b).natTrailingDegree := by
apply nilRank_eq_polyCharpoly_natTrailingDegree
| Mathlib/Algebra/Lie/Rank.lean | 64 | 66 |
/-
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.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... | (see `Relation.church_rosser`).
-/
def Join (r : α → α → Prop) : α → α → Prop := fun a b ↦ ∃ c, r a c ∧ r b c
section Join
open ReflTransGen ReflGen
| Mathlib/Logic/Relation.lean | 636 | 643 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | measure. -/
theorem measure_setLAverage_le_pos (hμ : μ s ≠ 0) (hs : NullMeasurableSet s μ)
| Mathlib/MeasureTheory/Integral/Average.lean | 629 | 630 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | rw [one_smul, one_mul]
| of _ =>
intros
rw [of_smul_def, prod_rcons, of.map_mul, mul_assoc, ← prod_rcons, ← equivPair_symm,
Equiv.symm_apply_apply]
| Mathlib/GroupTheory/CoprodI.lean | 588 | 592 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
/-!
This file establishes a `snoc : Vector α n → α → Vector α (n+1)` operation, that appends a single
element to the back of a vector.... | @[simp]
theorem mapAccumr₂_nil {f : α → β → σ → σ × φ} :
| Mathlib/Data/Vector/Snoc.lean | 146 | 147 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
/-!
# Properties of pointwise scalar multiplication of se... | -- This is also true for `ℚ`-normed spaces
theorem disjoint_ball_ball_iff (hδ : 0 < δ) (hε : 0 < ε) :
Disjoint (ball x δ) (ball y ε) ↔ δ + ε ≤ dist x y := by
refine ⟨fun h => le_of_not_lt fun hxy => ?_, ball_disjoint_ball⟩
rw [add_comm] at hxy
| Mathlib/Analysis/NormedSpace/Pointwise.lean | 197 | 201 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... |
theorem card_eq_one : #s = 1 ↔ ∃ a, s = {a} := by
| Mathlib/Data/Finset/Card.lean | 590 | 592 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | variable [MeasurableSingletonClass α]
section aux
omit [TopologicalSpace β]
/-- Auxiliary definition for `StronglyMeasurable.of_discrete`. -/
private noncomputable def simpleFuncAux (f : α → β) (g : ℕ → α) : ℕ → SimpleFunc α β
| 0 => .const _ (f (g 0))
| n + 1 => .piecewise {g n} (.singleton _) (.const _ <| f (g ... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 117 | 125 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Ker... | HasColimit.mk
{ cocone := coconeRightOpOfCone (limit.cone F)
isColimit := isColimitCoconeRightOpOfCone _ (limit.isLimit _) }
instance hasColimit_unop_of_hasLimit (F : Jᵒᵖ ⥤ Cᵒᵖ) [HasLimit F] : HasColimit F.unop :=
HasColimit.mk
{ cocone := coconeUnopOfCone (limit.cone F)
isColimit := isColimitC... | Mathlib/CategoryTheory/Limits/Opposites.lean | 406 | 421 |
/-
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.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | Mathlib/Data/Real/ConjExponents.lean | 204 | 204 | |
/-
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.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... | refine IH _ _ _ ?_ (insertRec H0 IH _)
exact not_mem_keys_of_nodupKeys_cons h
| Mathlib/Data/List/AList.lean | 327 | 328 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Moritz Doll
-/
import Mathlib.Algebra.GroupWithZero.Action.Opposite
import Mathlib.LinearAlgebra.Finsupp.VectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
im... | variable [AddCommMonoid M₁] [Module R M₁] [AddCommMonoid M₂] [Module R M₂] [AddCommMonoid N₂]
[Module R N₂]
variable [DecidableEq n] [Fintype n]
| Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 319 | 321 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Fin.Tuple.Reflection
/-!
# Lemmas for concrete matrices `Matrix (Fin m) (Fin n) α`
This file contains alternative ... | exact Iff.symm Fin.forall_fin_succ_pi
example (P : Matrix (Fin 2) (Fin 3) α → Prop) :
(∀ x, P x) ↔ ∀ a b c d e f, P !![a, b, c; d, e, f] :=
(forall_iff _).symm
| Mathlib/Data/Matrix/Reflection.lean | 56 | 60 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.CategoryTheory.Limits.FintypeCat
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.Ca... | the natural action of `Aut X` on `F.obj X` is transitive. -/
theorem isGalois_iff_pretransitive (X : C) [IsConnected X] :
IsGalois X ↔ MulAction.IsPretransitive (Aut X) (F.obj X) := by
rw [isGalois_iff_aux, Equiv.nonempty_congr <| quotientByAutTerminalEquivUniqueQuotient F X]
| Mathlib/CategoryTheory/Galois/GaloisObjects.lean | 81 | 84 |
/-
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... |
lemma rpow_intCast_mul (x : ℝ≥0∞) (n : ℤ) (z : ℝ) : x ^ (n * z) = (x ^ n) ^ z := by
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 920 | 921 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 483 | 484 | |
/-
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.Shift.CommShift
/-!
# Functors from a category to a category with a shift
Given a category `C`, and a category `D` equipped with a shift by a mo... | lemma shiftIso_add_inv_app (n m a a' a'' : A) (ha' : n + a = a') (ha'' : m + a' = a'') (X : C) :
(F.shiftIso (m + n) a a'' (by rw [add_assoc, ha', ha''])).inv.app X =
(F.shiftIso n a a' ha').inv.app X ≫
((F.shiftIso m a' a'' ha'').inv.app X)⟦n⟧' ≫
(shiftFunctorAdd D m n).inv.app ((F.functor a'').o... | Mathlib/CategoryTheory/Shift/SingleFunctors.lean | 63 | 68 |
/-
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.List.TakeDrop
import Mathlib.Data.List.Induction
/-!
# Prefixes, suffixes, infixes
This file proves properties about
* `List.isPrefix`: `l₁` is ... | Mathlib/Data/List/Infix.lean | 524 | 527 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | /-- If `s i` is a family of submodules, each is in its module,
then the closure of their span in the indexed product of the modules
is the product of their closures.
In case of a finite index type, this statement immediately follows from `Submodule.iSup_map_single`.
However, the statement is true for an infinite index... | Mathlib/Topology/Algebra/Module/Basic.lean | 224 | 236 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... |
@[simp]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 311 | 312 |
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