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) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Basic
/-!
# Isomorphisms about functors which preserve (co)limits
If `G` preserves limits, and `C` and `D` ... | ((isLimitOfPreserves G t).uniq (G.mapCone c₂) _ (by simp [← G.map_comp])).symm
variable [HasLimit F]
| Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 42 | 44 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | than `a`. If `a` is minimal, then `pred a = a`. -/
| Mathlib/Order/SuccPred/Basic.lean | 563 | 563 |
/-
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 | 1,835 | 1,838 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Data.Nat.Basic
/-!
# Cast of integers (additional theorems)
This ... |
section zsmul
| Mathlib/Data/Int/Cast/Basic.lean | 123 | 124 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,940 | 2,947 | |
/-
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.MeasureTheory.Integral.Bochner.Set
import Mathlib.Tactic.Generalize
/-!
# McShane integrability vs Bochner... | exact HasIntegral.unique (IntegrableOn.hasBoxIntegral this ⊥ rfl) (HasIntegral.mono hy bot_le)
/-- If `f : ℝⁿ → E` is a.e. continuous and bounded on a rectangular box `I`, then it is Box
integrable on `I` w.r.t. a locally finite measure `μ` with the same integral. -/
theorem AEContinuous.hasBoxIntegral [Complete... | Mathlib/Analysis/BoxIntegral/Integrability.lean | 310 | 318 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... |
/-- Rotating the first of two vectors by `θ` scales their Kahler form by `cos (-θ) + sin (-θ) * I`.
-/
theorem kahler_rotation_left' (x y : V) (θ : Real.Angle) :
o.kahler (o.rotation θ x) y = (-θ).toCircle * o.kahler x y := by
simp only [Real.Angle.toCircle_neg, Circle.coe_inv_eq_conj, kahler_rotation_left]
/--... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 180 | 188 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.PurelyInseparable.Basic
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts... | exact fun ⟨n, _⟩ ↦ h.iterate n (by rwa [← coe_iterateFrobenius, map_zero])
theorem PerfectRing.pNilradical_eq_bot (R : Type*) [CommSemiring R] (p : ℕ) [ExpChar R p]
[PerfectRing R p] : pNilradical R p = ⊥ :=
| Mathlib/FieldTheory/IsPerfectClosure.lean | 120 | 123 |
/-
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... |
theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by
funext a d
| Mathlib/Logic/Relation.lean | 149 | 151 |
/-
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... | (Eventually.of_forall (fun x ↦ ?_))
· apply (Integrable.sub h''i _).norm.const_mul
change IntegrableOn (fun _ ↦ c) (a i) μ
simp [integrableOn_const, mu_ai]
| Mathlib/MeasureTheory/Integral/Average.lean | 809 | 812 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... |
theorem image_sub_const_Ico (h : c ≤ a) :
((Ico a b).image fun x => x - c) = Ico (a - c) (b - c) := by
| Mathlib/Order/Interval/Finset/Nat.lean | 158 | 160 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... |
theorem dense_iff {s : Set α} : Dense s ↔ ∀ x, ∀ r > 0, (ball x r ∩ s).Nonempty :=
forall_congr' fun x => by
simp only [mem_closure_iff, Set.Nonempty, exists_prop, mem_inter_iff, mem_ball', and_comm]
theorem dense_iff_iUnion_ball (s : Set α) : Dense s ↔ ∀ r > 0, ⋃ c ∈ s, ball c r = univ := by
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,147 | 1,152 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | theorem C_mul' (a : R) (f : R[X]) : C a * f = a • f := by
| Mathlib/Algebra/Polynomial/Coeff.lean | 154 | 154 |
/-
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... | ⟨fun h f g hf hg hp => (h.2 hp.symm).imp hf.eq_one_of_isUnit hg.eq_one_of_isUnit, fun h =>
⟨hp1 ∘ hp.eq_one_of_isUnit, fun f g hfg =>
(h (g * C f.leadingCoeff) (f * C g.leadingCoeff) ?_ ?_ ?_).symm.imp
(isUnit_of_mul_eq_one f _)
(isUnit_of_mul_eq_one g _)⟩⟩
· rwa [Monic, leadingC... | Mathlib/Algebra/Polynomial/Monic.lean | 292 | 303 |
/-
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.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | rw [← Cochain.comp_assoc_of_second_is_zero_cochain, inl_snd, Cochain.zero_comp]
@[simp]
lemma inr_fst_assoc {K : CochainComplex C ℤ} {d e f : ℤ} (γ : Cochain F K d)
(he : 1 + d = e) (hf : 0 + e = f) :
(Cochain.ofHom (inr φ)).comp ((fst φ).1.comp γ he) hf = 0 := by
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 134 | 139 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | Mathlib/Topology/Connected/Basic.lean | 1,085 | 1,107 | |
/-
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... |
theorem supDegree_sum_le {ι} {s : Finset ι} {f : ι → R[A]} :
(∑ i ∈ s, f i).supDegree D ≤ s.sup (fun i => (f i).supDegree D) := by
| Mathlib/Algebra/MonoidAlgebra/Degree.lean | 240 | 242 |
/-
Copyright (c) 2020 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Johan Commelin
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.... | theorem zeroLocus_vanishingIdeal_eq_closure (t : Set (ProjectiveSpectrum 𝒜)) :
zeroLocus 𝒜 (vanishingIdeal t : Set A) = closure t := by
apply Set.Subset.antisymm
· rintro x hx t' ⟨ht', ht⟩
obtain ⟨fs, rfl⟩ : ∃ s, t' = zeroLocus 𝒜 s := by rwa [isClosed_iff_zeroLocus] at ht'
rw [subset_zeroLocus_iff_su... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 307 | 314 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | @[simp]
theorem cylinder_univ (s : Finset ι) : cylinder s (univ : Set (∀ i : s, α i)) = univ := by
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 165 | 166 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Disjointed
import Mathlib.Algebra.Order.Ring.Prod
import Mathlib.Data.Int.Interval
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!... | Mathlib/Order/Interval/Finset/Box.lean | 35 | 35 | |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | normEDS_four ..
lemma ψ_even_ofNat (m : ℕ) : W.ψ (2 * (m + 3)) * W.ψ₂ =
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 465 | 467 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | refine ⟨max C 1, lt_max_of_lt_right one_pos, fun i => le_trans (eLpNorm_mono fun x => ?_) (hCg i)⟩
rw [norm_indicator_eq_indicator_norm, norm_indicator_eq_indicator_norm]
exact Set.indicator_le_indicator_of_subset
(fun x hx => Set.mem_setOf_eq ▸ le_trans (le_max_left _ _) hx) (fun _ => norm_nonneg _) _
end U... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 699 | 721 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | simp only [realize, max, realize_not, eq_iff_iff]
tauto
| Mathlib/ModelTheory/Semantics.lean | 294 | 295 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.CategoryTheory.ConcreteCategory.Bundled
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheor... | (α_ F G H).hom.app X = eqToHom (by simp) :=
rfl
| Mathlib/CategoryTheory/Category/Cat.lean | 129 | 130 |
/-
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... | @[simp]
theorem mem_filterMap {b : β} : b ∈ s.filterMap f f_inj ↔ ∃ a ∈ s, f a = some b :=
| Mathlib/Data/Finset/Image.lean | 542 | 543 |
/-
Copyright (c) 2023 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Affine.AddTorsorBases
/-!
# Intrinsic frontier and interior
This file defines the intrinsic frontier, interior and closur... | · rintro _ ⟨t, ht, rfl⟩ hx
obtain ⟨y, hyt, hys⟩ := h _ ht hx
exact ⟨⟨_, subset_affineSpan 𝕜 s hys⟩, hyt, hys⟩
· by_contra hc
obtain ⟨z, hz₁, hz₂⟩ := h _ (affineSpan 𝕜 s).closed_of_finiteDimensional.isOpen_compl hc
exact hz₁ (subset_affineSpan 𝕜 s hz₂)
variable {𝕜}
@[simp]
theorem closure_diff_... | Mathlib/Analysis/Convex/Intrinsic.lean | 267 | 277 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 461 | 463 | |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | @[simp]
theorem stdBasisMatrix_mul_mul_stdBasisMatrix [Fintype n]
(i : l) (i' : m) (j' : n) (j : o) (a : α) (x : Matrix m n α) (b : α) :
stdBasisMatrix i i' a * x * stdBasisMatrix j' j b = stdBasisMatrix i j (a * x i' j' * b) := by
| Mathlib/Data/Matrix/Basis.lean | 236 | 239 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | ∃ m' n' : ℤ, gcd m' n' = 1 ∧ m = m' * gcd m n ∧ n = n' * gcd m n :=
⟨_, _, gcd_div_gcd_div_gcd H, (Int.ediv_mul_cancel gcd_dvd_left).symm,
(Int.ediv_mul_cancel gcd_dvd_right).symm⟩
| Mathlib/Data/Int/GCD.lean | 275 | 278 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | have inv : Function.LeftInverse inv_fun to_fun ∧ Function.RightInverse inv_fun to_fun :=
if hmn0 : m * n = 0 then by
rcases h.eq_of_mul_eq_zero hmn0 with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
· constructor
· intro x; rfl
| Mathlib/Data/ZMod/Basic.lean | 851 | 855 |
/-
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.Algebra.MonoidAlgebra.Defs
/-!
# Division of `AddMonoidAlgebra` by monomials
This file is most important for when `G = ℕ` (polynomials) or `G = σ →₀ ℕ` (mu... | theorem of'_dvd_iff_modOf_eq_zero [IsCancelAdd G] {x : k[G]} {g : G} :
of' k G g ∣ x ↔ x %ᵒᶠ g = 0 := by
constructor
· rintro ⟨x, rfl⟩
rw [of'_mul_modOf]
· intro h
rw [← divOf_add_modOf x g, h, add_zero]
exact dvd_mul_right _ _
end
end AddMonoidAlgebra
| Mathlib/Algebra/MonoidAlgebra/Division.lean | 175 | 186 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | /-- To bound the Hausdorff measure (or, more generally, for a measure defined using
`MeasureTheory.Measure.mkMetric`) of a set, one may use coverings with maximum diameter tending to
`0`, indexed by any sequence of countable types. -/
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 494 | 496 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... |
@[simp]
nonrec theorem volume_pi_closedBall (a : ι → ℝ) {r : ℝ} (hr : 0 ≤ r) :
volume (Metric.closedBall a r) = ENNReal.ofReal ((2 * r) ^ Fintype.card ι) := by
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 264 | 267 |
/-
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.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Convex.Deriv
/-!
# The function `x ↦ ... | simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
rw [deriv2_mul_log]
| Mathlib/Analysis/SpecialFunctions/Log/NegMulLog.lean | 125 | 126 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | rw [closure_singleton_eq, AddMonoidHom.mem_mrange]; rfl
theorem closure_singleton_zero : closure ({0} : Set A) = ⊥ := by
simp [eq_bot_iff_forall, mem_closure_singleton, nsmul_zero]
/-- The additive submonoid generated by an element. -/
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 454 | 459 |
/-
Copyright (c) 2020 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.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | ENNReal.ofReal_eq_zero, if_false, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm]
have h_rpow : essSup (‖‖f ·‖ ^ q‖ₑ) μ = essSup (‖f ·‖ₑ ^ q) μ := by
congr
ext1 x
conv_rhs => rw [← enorm_norm]
rw [← Real.enorm_rpow_of_nonneg (norm_nonneg _) hq_pos.le]
rw [h_rpow]
have h_rp... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 509 | 519 |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... |
instance PosSMulReflectLT.toPosSMulReflectLE [PosSMulReflectLT α β] : PosSMulReflectLE α β :=
| Mathlib/Algebra/Order/Module/Defs.lean | 690 | 691 |
/-
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
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... |
@[simp]
| Mathlib/Order/Interval/Set/ProjIcc.lean | 77 | 78 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
/-!
# Proba... | HasPDF X ℙ μ ↔ (map X ℙ).HaveLebesgueDecomposition μ ∧ map X ℙ ≪ μ := by
rw [hasPDF_iff]
simp only [hX, true_and]
variable (X ℙ μ) in
| Mathlib/Probability/Density.lean | 82 | 86 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | theorem Rel.trans {x y z : α × α} (a : Rel α x y) (b : Rel α y z) : Rel α x z := by
| Mathlib/Data/Sym/Sym2.lean | 69 | 69 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | irreducible_def volumeForm : E [⋀^Fin n]→ₗ[ℝ] ℝ := by
classical
cases n with
| zero =>
let opos : E [⋀^Fin 0]→ₗ[ℝ] ℝ := .constOfIsEmpty ℝ E (Fin 0) (1 : ℝ)
exact o.eq_or_eq_neg_of_isEmpty.by_cases (fun _ => opos) fun _ => -opos
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 166 | 171 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
/-!
# Some results on the rank... | end
section
| Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 51 | 53 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 805 | 809 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... |
This typeclass describes a situation in which a measure `μ` on `α ⧸ G` can be computed by
taking a measure `ν` on `α` of the intersection of the pullback with a fundamental domain.
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 710 | 712 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
/-!
# The Karoubi envelope ... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 303 | 305 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.CauSeq.Completion
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Cast.Defs
/-!
# Real numbers from Cauc... | induction a using Real.ind_mk
| Mathlib/Data/Real/Basic.lean | 323 | 323 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | Mathlib/Data/Int/GCD.lean | 424 | 426 | |
/-
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.Category.ModuleCat.Adjunctions
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.Al... | theorem homEquiv_def (A B C : Rep k G) : (ihom.adjunction A).homEquiv B C = Rep.homEquiv A B C :=
congrFun (congrFun (Adjunction.mkOfHomEquiv_homEquiv _) _) _
@[simp]
theorem ihom_ev_app_hom (A B : Rep k G) :
Action.Hom.hom ((ihom.ev A).app B) = ModuleCat.ofHom
| Mathlib/RepresentationTheory/Rep.lean | 376 | 381 |
/-
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.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
/-!
# Adhesive c... | theorem IsPushout.IsVanKampen.mono_of_mono_right [Mono g] {H : IsPushout f g h i}
(H' : H.IsVanKampen) : Mono h :=
IsKernelPair.mono_of_isIso_fst
((H' f (𝟙 _) (𝟙 _) f (𝟙 _) (𝟙 _) g h (IsPullback.of_vert_isIso ⟨by simp⟩)
(IsKernelPair.id_of_mono g) ⟨rfl⟩ H.1 ⟨by simp⟩).mp
(IsPushout.of_vert_i... | Mathlib/CategoryTheory/Adhesive.lean | 209 | 214 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | conv_rhs =>
rw [← modByMonic_add_div p this,
(modByMonic_eq_zero_iff_dvd this).2 (pow_rootMultiplicity_dvd _ _)]
simp
theorem exists_eq_pow_rootMultiplicity_mul_and_not_dvd (p : R[X]) (hp : p ≠ 0) (a : R) :
∃ q : R[X], p = (X - C a) ^ p.rootMultiplicity a * q ∧ ¬ (X - C a) ∣ q := by
classical
... | Mathlib/Algebra/Polynomial/Div.lean | 557 | 566 |
/-
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... | · intro x hx
rw [pow_succ, mul_apply, pow_succ, mul_apply, h _ hx, hk]
rwa [mem_inter, apply_mem_support, ← h _ hx, apply_mem_support, ← mem_inter]
theorem disjoint_iff_disjoint_support : Disjoint f g ↔ _root_.Disjoint f.support g.support := by
simp [disjoint_iff_eq_or_eq, disjoint_iff, disjoint_iff, Finse... | Mathlib/GroupTheory/Perm/Support.lean | 405 | 410 |
/-
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... |
end Tree
| Mathlib/Combinatorics/Enumerative/Catalan.lean | 191 | 194 |
/-
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.Order.Bounds.Basic
/-!
# Intervals in Lattices
In this file, we provide instances of lattice structures on intervals within lattices.
Some of them de... | rw [_root_.isCompl_iff, Iic.disjoint_iff, Iic.codisjoint_iff]
protected lemma complementedLattice_iff [Lattice α] [OrderBot α] :
| Mathlib/Order/LatticeIntervals.lean | 127 | 129 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | P.comapOfInjective N.subtype Subtype.coe_injective (by rwa [range_subtype])
@[simp]
theorem coe_subtype (h : P ≤ N) : P.subtype h = subgroupOf P N :=
| Mathlib/GroupTheory/Sylow.lean | 138 | 141 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | theorem injOn_tan : InjOn tan (Ioo (-(π / 2)) (π / 2)) :=
strictMonoOn_tan.injOn
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 900 | 902 |
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | rfl
theorem isUnit_T (n : ℤ) : IsUnit (T n : R[T;T⁻¹]) :=
| Mathlib/Algebra/Polynomial/Laurent.lean | 227 | 229 |
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.... | def skewAdjoint.negISMul : skewAdjoint A →ₗ[ℝ] selfAdjoint A where
| Mathlib/Data/Complex/Module.lean | 334 | 334 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Control.EquivFunctor
import Mathlib.Data.Option.Basic
import Mathlib.Data.Subtype
import Mathlib.Logic.Equiv.Defs
/-!
# Equivalences for `Option α`
We def... | def optionIsSomeEquiv (α) : { x : Option α // x.isSome } ≃ α where
toFun o := Option.get _ o.2
| Mathlib/Logic/Equiv/Option.lean | 277 | 278 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | /-- An `n`-Freiman homomorphism preserves products of `n` elements. -/
map_prod_eq_map_prod ⦃s t : Multiset α⦄ (hsA : ∀ ⦃x⦄, x ∈ s → x ∈ A) (htA : ∀ ⦃x⦄, x ∈ t → x ∈ A)
(hs : Multiset.card s = n) (ht : Multiset.card t = n) :
(s.map f).prod = (t.map f).prod ↔ s.prod = t.prod
@[to_additive]
| Mathlib/Combinatorics/Additive/FreimanHom.lean | 109 | 114 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | theorem lift_comp_kernelIsoOfEq_inv {Z} {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g)
(e : Z ⟶ X) (he) :
kernel.lift _ e he ≫ (kernelIsoOfEq h).inv = kernel.lift _ e (by simp [h, he]) := by
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 388 | 390 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | theorem MeasurableSet.map_coe_volume {s : Set α} (hs : MeasurableSet s) :
volume.map ((↑) : s → α) = restrict volume s := by
rw [volume_set_coe_def, (MeasurableEmbedding.subtype_coe hs).map_comap volume, Subtype.range_coe]
theorem volume_image_subtype_coe {s : Set α} (hs : MeasurableSet s) (t : Set s) :
volu... | Mathlib/MeasureTheory/Measure/Restrict.lean | 861 | 866 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | theorem cons_get_inits_core :
∀ (a : α) (n : ℕ) (l : List α) (s : Stream' α),
| Mathlib/Data/Stream/Init.lean | 665 | 666 |
/-
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.Data.Finite.Sum
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Ideal.Quotient.Operations
imp... | * `Q`, for ring homs `(R : Type u) → (S : Type v)`.
-/
lemma polynomial_induction
(P : ∀ (R : Type u) [CommRing R] (S : Type u) [CommRing S], (R →+* S) → Prop)
(Q : ∀ (R : Type u) [CommRing R] (S : Type v) [CommRing S], (R →+* S) → Prop)
(polynomial : ∀ (R) [CommRing R], P R R[X] C)
(fg_ker : ∀ (R : Typ... | Mathlib/RingTheory/FinitePresentation.lean | 461 | 471 |
/-
Copyright (c) 2022 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Order.Filter.Lift
import Mathlib.Order.Filter.AtTopBot.Basic
/-!
# The filter of small sets
This file defines the ... |
end Filter
| Mathlib/Order/Filter/SmallSets.lean | 182 | 184 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | theorem spanningCoe_top : (⊤ : Subgraph G).spanningCoe = G := rfl
@[simp]
theorem spanningCoe_bot : (⊥ : Subgraph G).spanningCoe = ⊥ := rfl
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 555 | 559 |
/-
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... |
@[to_additive ne_zero_of_mem_unit_sphere]
| Mathlib/Analysis/Normed/Group/Basic.lean | 632 | 633 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | simp only [smul_mk, of, RingHom.comp_apply, ← (mk f).map_mul, Polynomial.nnqsmul_eq_C_mul]
qsmul_def q x :=
-- Porting note: I gave the explicit motive and changed `rw` to `simp`.
AdjoinRoot.induction_on f (C := fun y ↦ q • y = (of f) q * y) x fun p ↦ by
simp only [smul_mk, of, RingHom.comp_apply,... | Mathlib/RingTheory/AdjoinRoot.lean | 355 | 361 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... |
/--
The maximum degree of a nonempty graph is less than the number of vertices. Note that the assumption
that `V` is nonempty is necessary, as otherwise this would assert the existence of a
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 374 | 377 |
/-
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 | 1,667 | 1,668 | |
/-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
/-!
# Bivariate polynomials
This file introduces the notation `R[X][Y]` for the polynomial ring `R[X][X]` in two variables,
and the notat... | lemma evalEval_neg (x y : R) (p : R[X][Y]) : (-p).evalEval x y = -p.evalEval x y := by
simp only [evalEval, eval_neg]
@[simp]
lemma evalEval_sub (x y : R) (p q : R[X][Y]) :
(p - q).evalEval x y = p.evalEval x y - q.evalEval x y := by
| Mathlib/Algebra/Polynomial/Bivariate.lean | 94 | 99 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
/-!... |
@[simp]
| Mathlib/Data/Rat/Sqrt.lean | 47 | 48 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Aesop
import Mathlib.Order.BoundedOrder.Lattice
/-!
# Disjointness and complements
This file defines `Disjoint`, `Codisjoint`, and the `IsCompl` predicate.
... | Mathlib/Order/Disjoint.lean | 819 | 820 | |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Mathlib/Data/Option/Basic.lean | 414 | 417 | |
/-
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
-/
import Mathlib.CategoryTheory.Category.Basic
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Tactic.CategoryTheory.CheckComposi... | Mathlib/CategoryTheory/Functor/Basic.lean | 136 | 139 | |
/-
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.InducedShiftSequence
import Mathlib.CategoryTheory.Shift.Localization
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
import Mathlib.... | ext <;> dsimp <;> simp [one_smul, shiftFunctorZero_hom_app_f]
lemma shiftShortComplexFunctorIso_add'_hom_app
(n m mn : ℤ) (hmn : m + n = mn) (a a' a'' : ℤ) (ha' : n + a = a') (ha'' : m + a' = a'')
(K : CochainComplex C ℤ) :
(shiftShortComplexFunctorIso C mn a a'' (by rw [← ha'', ← ha', ← add_assoc, hmn])... | Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean | 60 | 70 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,054 | 2,057 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 248 | 248 | |
/-
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, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | theorem div_eq_of_eq_mul' {a b c : G} (h : a = b * c) : a / b = c := by
rw [h, div_eq_mul_inv, mul_comm, inv_mul_cancel_left]
| Mathlib/Algebra/Group/Basic.lean | 926 | 927 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Lebesgu... |
end
variable (μ)
theorem measure_closedBall_le_integral : μ.real (closedBall c f.rIn) ≤ ∫ x, f x ∂μ := by calc
μ.real (closedBall c f.rIn) = ∫ x in closedBall c f.rIn, 1 ∂μ := by simp
_ = ∫ x in closedBall c f.rIn, f x ∂μ := setIntegral_congr_fun measurableSet_closedBall
(fun x hx ↦ (one_of_mem_closedBal... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 93 | 101 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | theorem card_eq_coe_fintype_card [Fintype α] : card α = Fintype.card α := by
simp [card]
@[simp high]
| Mathlib/SetTheory/Cardinal/Finite.lean | 272 | 275 |
/-
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.Algebra.Polynomial.Eval.Defs
import Mathlib.Analysis.Asymptotics.Lemmas
/-!
# Super-Polynomial Function Decay
This file defines a predicate `Asymptotics.Supe... | theorem SuperpolynomialDecay.mul [ContinuousMul β] (hf : SuperpolynomialDecay l k f)
(hg : SuperpolynomialDecay l k g) : SuperpolynomialDecay l k (f * g) := fun z => by
| Mathlib/Analysis/Asymptotics/SuperpolynomialDecay.lean | 80 | 81 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 980 | 981 | |
/-
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 | 1,574 | 1,575 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | · rintro s t hst ⟨V, hV, hV'⟩
exact ⟨V, hV, (mul_subset_mul_right hst).trans hV'⟩
· rintro s t ⟨V, V_in, hV'⟩ ⟨W, W_in, hW'⟩
use V ∩ W, inter_mem V_in W_in
rw [union_mul]
exact
union_subset ((mul_subset_mul_left V.inter_subset_left).trans hV')
((mul_subset_mul_left V.inter_subset_right... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,070 | 1,085 |
/-
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.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
/-!
# Semisimple Lie algebras
The famous Cartan-Dynkin-Killing classification of semisim... | · exact h₁ _ <| LieAlgebra.ofAbelianIsSolvable I
· rw [← abelian_of_solvable_ideal_eq_bot_iff]
exact h₁ _ <| abelian_derivedAbelianOfIdeal I
namespace IsSimple
variable [IsSimple R L]
| Mathlib/Algebra/Lie/Semisimple/Basic.lean | 71 | 77 |
/-
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.Topology.ContinuousMap.Bounded.ArzelaAscoli
import Mathlib.Topology.ContinuousMap.Bounded.Normed
import Mathlib.Topology.MetricSpace.Gluing
impor... | let ⟨_, Z2⟩ := Classical.choose_spec (exists_minimizer X Y)
Z2 g hg
| Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean | 428 | 430 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComple... | simp only [N₁Γ₀_app]
rfl
theorem N₁Γ₀_inv_app (K : ChainComplex C ℕ) :
N₁Γ₀.inv.app K = (toKaroubi _).map (Γ₀NondegComplexIso K).inv ≫
(Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.hom := by
| Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 86 | 91 |
/-
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.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | le_max_iff, mul_comm y.num]
exact natDegree_add_le _ _
end IntDegree
end RatFunc
| Mathlib/FieldTheory/RatFunc/Degree.lean | 110 | 121 |
/-
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... | ((Nat.dvd_add_iff_right (Nat.dvd_mul_left m (n * x))).mpr
((congr_arg _ h).mpr (Nat.dvd_mul_right m n)))
| Mathlib/Data/Nat/GCD/Basic.lean | 239 | 240 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Mul
import Mathlib.Analysis.Convex.SpecificFunctions.Basic
imp... | (convexOn_rpow hp).map_sum_le hw hw' hz
theorem arith_mean_le_rpow_mean (w z : ι → ℝ) (hw : ∀ i ∈ s, 0 ≤ w i) (hw' : ∑ i ∈ s, w i = 1)
(hz : ∀ i ∈ s, 0 ≤ z i) {p : ℝ} (hp : 1 ≤ p) :
∑ i ∈ s, w i * z i ≤ (∑ i ∈ s, w i * z i ^ p) ^ (1 / p) := by
have : 0 < p := by positivity
rw [← rpow_le_rpow_iff _ _ this... | Mathlib/Analysis/MeanInequalitiesPow.lean | 72 | 86 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
import Mathlib.Analysis.SpecialFunctions... | convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1
rw_mod_cast [← sin_sq_add_cos_sq x]
ring
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 25 | 28 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,368 | 1,368 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.Seminorm
import Mathlib.Data.Real.Sqrt
import Mathlib.Topology.Algebra.Equicontinuit... | rw [hp.tendsto_nhds u y₀]
exact forall₃_congr fun _ _ _ => Filter.eventually_atTop
end Tendsto
| Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 380 | 383 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... |
@[simp]
theorem toIcoDiv_add_right' (a b : α) : toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1 := by
simpa only [one_zsmul] using toIcoDiv_add_zsmul' hp a b 1
| Mathlib/Algebra/Order/ToIntervalMod.lean | 253 | 257 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,395 | 1,396 | |
/-
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.... |
@[simp]
theorem cast_inj (eq : n = m) {a b : Fin n} : a.cast eq = b.cast eq ↔ a = b := by
simp [← val_inj]
@[simp]
| Mathlib/Data/Fin/Basic.lean | 537 | 542 |
/-
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 ... | l₁.flatMap f <:+ l₂.flatMap f :=
(h.map _).flatten
@[gcongr]
| Mathlib/Data/List/Infix.lean | 73 | 76 |
/-
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 ... | gdiff.continuousOn hi
end
/-!
| Mathlib/Analysis/Calculus/MeanValue.lean | 394 | 398 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | refine fun z ↦ ⟨?_, ?_, ?_⟩ <;>
refine fun h ↦ ih' (Args.P24 _ _ _) (TransGen.single ?_)
· exact (cutExpand_add_right {y}).2 (cutExpand_pair_left h)
· exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_left h)
· exact (cutExpand_add_left {x₁}).2 (cutExpand_pair_right h)
lemma ih₂₁ (ih' : ∀ a, ArgsRel a (Arg... | Mathlib/SetTheory/Surreal/Multiplication.lean | 295 | 301 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,798 | 2,804 |
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