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/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
/-!
# Integration by parts for line derivatives
Let ... | variable [BorelSpace E]
lemma integral_bilinear_hasLineDerivAt_right_eq_neg_left_of_integrable_aux2
[FiniteDimensional ℝ E] {μ : Measure (E × ℝ)} [IsAddHaarMeasure μ]
{f f' : E × ℝ → F} {g g' : E × ℝ → G} {B : F →L[ℝ] G →L[ℝ] W}
(hf'g : Integrable (fun x ↦ B (f' x) (g x)) μ)
(hfg' : Integrable (fun x ↦... | Mathlib/Analysis/Calculus/LineDeriv/IntegrationByParts.lean | 75 | 93 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... | have hab := mul_ne_zero ha hb
by_cases hF : ringChar F = 2
·-- case `ringChar F = 2`
rw [quadraticCharFun_eq_one_of_char_two hF ha, quadraticCharFun_eq_one_of_char_two hF hb,
quadraticCharFun_eq_one_of_char_two hF hab, mul_one]
· -- case of odd characteristic
rw [quadraticCharFun_eq_pow_of_char_ne... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 100 | 122 |
/-
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.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functor... | theorem Pi.multipliable {f : ι → ∀ x, π x} : Multipliable f ↔ ∀ x, Multipliable fun i ↦ f i x := by
simp only [Multipliable, Pi.hasProd, Classical.skolem]
| Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 314 | 315 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
/-!
# Filters used in box-based integrals
First ... | wlog hc : c₁ ≤ c₂ with H
· simpa [hU, _root_.and_comm] using
@H _ _ I c₂ c₁ l r₂ r₁ π₂ π₁ h₂ h₁ hU.symm (le_of_not_le hc)
| Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 347 | 349 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Rat
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.Rat.Sqrt
imp... | @[simp]
theorem Int.not_irrational (m : ℤ) : ¬Irrational m := fun h => h.ne_int m rfl
| Mathlib/Data/Real/Irrational.lean | 192 | 194 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 184 | 197 | |
/-
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... | converge in Lp to `g`. -/
theorem tendsto_Lp_finite_of_tendstoInMeasure [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : MemLp g p μ) (hui : UnifIntegrable f p μ)
(hfg : TendstoInMeasure μ f atTop g) : Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0) := by
refi... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 593 | 601 |
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
/-!
# ω-limits
For a function `ϕ : τ → α → β` where `β` is a topological space, we
define the ω-limit under `ϕ` of... | subset_iInter fun _i ↦ omegaLimit_mono_right _ _ (iInter_subset _ _)
theorem omegaLimit_union : ω f ϕ (s₁ ∪ s₂) = ω f ϕ s₁ ∪ ω f ϕ s₂ := by
| Mathlib/Dynamics/OmegaLimit.lean | 150 | 152 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
/-!
# Specific subobjects
We define `equalizerSubobject`, `kernelSubobject` and `imageSubobject`, which ar... | Mathlib/CategoryTheory/Subobject/Limits.lean | 463 | 467 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | theorem _root_.IsOpen.exists_lt_isCompact [Regular μ] ⦃U : Set α⦄ (hU : IsOpen U) {r : ℝ≥0∞}
(hr : r < μ U) : ∃ K, K ⊆ U ∧ IsCompact K ∧ r < μ K :=
Regular.innerRegular hU r hr
/-- The measure of an open set is the supremum of the measures of compact sets it contains. -/
theorem _root_.IsOpen.measure_eq_iSup_isC... | Mathlib/MeasureTheory/Measure/Regular.lean | 983 | 989 |
/-
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.CartanSubalgebra
import Mathlib.Algebra.Lie.Rank
/-!
# Existence of Cartan subalgebras
In this file we prove existence of Cartan subalge... | lemma lieCharpoly_coeff_natDegree [Nontrivial R] (i j : ℕ) (hij : i + j = finrank R M) :
((lieCharpoly R M x y).coeff i).natDegree ≤ j := by
classical
rw [← mul_one j, lieCharpoly, coeff_map]
apply MvPolynomial.aeval_natDegree_le
· apply (polyCharpoly_coeff_isHomogeneous φ (chooseBasis R L) _ _ hij).totalDe... | Mathlib/Algebra/Lie/CartanExists.lean | 100 | 110 |
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 815 | 816 | |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,044 | 3,046 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
/-!
# Open immersions of structured spaces
We say that a m... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 1,275 | 1,280 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Int.Units
import M... | theorem normalize_of_nonneg {z : ℤ} (h : 0 ≤ z) : normalize z = z := by
rw [normalize_apply, normUnit_eq, if_pos h, Units.val_one, mul_one]
| Mathlib/Algebra/GCDMonoid/Nat.lean | 67 | 68 |
/-
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
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.ExistsOfLE
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Unbundled.Basic
impor... | tsub_eq_zero_of_le <| zero_le a
theorem tsub_self_add (a b : α) : a - (a + b) = 0 :=
tsub_eq_zero_of_le <| self_le_add_right _ _
theorem tsub_pos_iff_not_le : 0 < a - b ↔ ¬a ≤ b := by
rw [pos_iff_ne_zero, Ne, tsub_eq_zero_iff_le]
| Mathlib/Algebra/Order/Sub/Basic.lean | 45 | 51 |
/-
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.Functor.ReflectsIso.Basic
import Mathlib.CategoryTheory.MorphismProperty.Basic
/-!
# Morphism properties that are inverted by a functor
In this ... | rw [isIso_pi_iff]
intro j
exact hF j _ (hf j)
end IsInvertedBy
/-- The full subcategory of `C ⥤ D` consisting of functors inverting morphisms in `W` -/
def FunctorsInverting (W : MorphismProperty C) (D : Type*) [Category D] :=
ObjectProperty.FullSubcategory fun F : C ⥤ D => W.IsInvertedBy F
| Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 85 | 93 |
/-
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 | 1,153 | 1,154 | |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | @[deprecated (since := "2024-11-30")]
alias multiplicity.Finite.le_multiplicity_of_le_emultiplicity :=
FiniteMultiplicity.le_multiplicity_of_le_emultiplicity
theorem multiplicity_lt_of_emultiplicity_lt {n : ℕ} (h : emultiplicity a b < n) :
multiplicity a b < n := by
exact_mod_cast multiplicity_le_emultiplicity... | Mathlib/RingTheory/Multiplicity.lean | 168 | 178 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | map_valEmbedding_attachFin _
@[simp]
theorem map_valEmbedding_Ico : (Ico a b).map Fin.valEmbedding = Ico (a : ℕ) b :=
map_valEmbedding_attachFin _
@[simp]
theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc (a : ℕ) b :=
map_valEmbedding_attachFin _
@[simp]
theorem map_valEmbedding_Ioo : (Ioo a ... | Mathlib/Order/Interval/Finset/Fin.lean | 175 | 186 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | rw [@norm_sub_mul_self 𝕜]
ring
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 472 | 474 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | simp [Function.star_sumElim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
theorem schur_complement_eq₂₂ [Fintype m] [Fintype n] [DecidableEq n] (A : Matrix m m 𝕜)
| Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 464 | 469 |
/-
Copyright (c) 2024 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Topology.ContinuousMap.ZeroAtInfty
/-!
# ZeroAtInftyContinuousMapClass in normed additive groups
In this file we give a characterization of the predicate ... | theorem zero_at_infty_of_norm_le (f : E → F)
(h : ∀ (ε : ℝ) (_hε : 0 < ε), ∃ (r : ℝ), ∀ (x : E) (_hx : r < ‖x‖), ‖f x‖ < ε) :
Tendsto f (cocompact E) (𝓝 0) := by
rw [tendsto_zero_iff_norm_tendsto_zero]
intro s hs
rw [mem_map, Metric.mem_cocompact_iff_closedBall_compl_subset 0]
rw [Metric.mem_nhds_iff] ... | Mathlib/Analysis/Normed/Group/ZeroAtInfty.lean | 38 | 49 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.CategoryTheory.Sites.Canonical
/-!
# Grothendieck Topology and Sheaves on the Category of Types
In this file we define a Grothendieck topology on the category ... | rw [← op_comp]
--congr 2 -- Porting note: This tactic didn't work. Find an alternative.
suffices ((↾fun _ ↦ PUnit.unit) ≫ ↾fun _ ↦ f (Classical.choose hf)) = f by rw [this]
funext x
exact congr_arg f (Classical.choose_spec hf x).symm
/-- Given a sheaf `S`, construct an equivalence `S(α) ≃ (α → S(*))`. -/
@[s... | Mathlib/CategoryTheory/Sites/Types.lean | 108 | 117 |
/-
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... |
theorem pairwise_disjoint_powersetCard (s : Finset α) :
Pairwise fun i j => Disjoint (s.powersetCard i) (s.powersetCard j) := fun _i _j hij =>
Finset.disjoint_left.mpr fun _x hi hj =>
hij <| (mem_powersetCard.mp hi).2.symm.trans (mem_powersetCard.mp hj).2
theorem powerset_card_disjiUnion (s : Finset α) :
... | Mathlib/Data/Finset/Powerset.lean | 260 | 271 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | apply Nat.coprime_of_dvd
-- If `k` is a prime number that divides both `n` and `b`, then we know that `n = m * k` and
-- `b = j * k` for some natural numbers `m` and `j`. We substitute these into the hypothesis.
rintro k hk ⟨m, rfl⟩ ⟨j, rfl⟩
-- Because prime numbers do not divide 1, it suffices to s... | Mathlib/NumberTheory/FermatPsp.lean | 75 | 99 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Sheaf
/-!
# The plus construction for presheaves.
This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` i... | refine Multiequalizer.hom_ext _ _ _ (fun I => ?_)
erw [Multiequalizer.lift_ι]
simp only [unop_op, op_unop, diagram_map, Category.assoc, limit.lift_π,
Multifork.ofι_π_app]
let ee : (J.pullback (I.map e).f).obj S.unop ⟶ ⊤ := homOfLE (OrderTop.le_top _)
erw [← colimit.w _ ee.op, ι_colimMap_assoc, colimit.ι_p... | Mathlib/CategoryTheory/Sites/Plus.lean | 220 | 228 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | let g : Module.End R L := toEnd R H L y
have hfg : g.comp (H : Submodule R L).subtype = (H : Submodule R L).subtype.comp f := by
ext z
simp only [toEnd_apply_apply, Submodule.subtype_apply,
| Mathlib/Algebra/Lie/Weights/Cartan.lean | 175 | 178 |
/-
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.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
/-!
# Matrix results for barycentric co-ordinates
Results about the ... |
end CommRing
end AffineBasis
| Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 160 | 165 |
/-
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
/-... | right_inv _ := funext <| Sym2.ind fun _ _ => rfl
| Mathlib/Data/Sym/Sym2.lean | 188 | 189 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Limits
import Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
import Mathlib.CategoryTheory.Adhesive
import Mathlib.Categ... | -- evaluation preserves limits.
theorem liftToPlusObjLimitObj_fac {K : Type max v u} [SmallCategory K] [FinCategory K]
[HasLimitsOfShape K D] [PreservesLimitsOfShape K (forget D)]
[ReflectsLimitsOfShape K (forget D)] (F : K ⥤ Cᵒᵖ ⥤ D) (X : C)
(S : Cone (F ⋙ J.plusFunctor D ⋙ (evaluation Cᵒᵖ D).obj (op X))) ... | Mathlib/CategoryTheory/Sites/LeftExact.lean | 148 | 171 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | · obtain ⟨v, vs, xv, vu⟩ : ∃ v ∈ t, x ∈ v ∧ v ⊆ Sum.inr ⁻¹' u :=
ht.exists_subset_of_mem_open hxu (isOpen_sum_iff.1 u_open).2
exact ⟨Sum.inr '' v, mem_union_right _ (mem_image_of_mem _ vs), mem_image_of_mem _ xv,
image_subset_iff.2 vu⟩
| Mathlib/Topology/Bases.lean | 892 | 896 |
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Yaël Dillies, Jineon Baek
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebr... | [R. Taylor and A. Wiles, *Ring-theoretic properties of certain Hecke algebras*][Taylor-Wiles-FLT]).
An ongoing Lean formalisation of the proof, using mathlib as a dependency, is taking place at
| Mathlib/NumberTheory/FLT/Basic.lean | 39 | 40 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Fin.Tuple.Basic
/-!
# Matrix and vector notation
This file defines notation for vectors and matrices. Given `a b c d : α`,
the notation a... |
@[simp]
theorem head_cons (x : α) (u : Fin m → α) : vecHead (vecCons x u) = x :=
| Mathlib/Data/Fin/VecNotation.lean | 188 | 190 |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.... | {a n' d' : α} → IsRat a (.negOfNat n) d → n = n' → d = d' → a = -(n' / d')
| _, _, _, ⟨_, rfl⟩, rfl, rfl => by simp [div_eq_mul_inv]
| Mathlib/Tactic/NormNum/Result.lean | 228 | 230 |
/-
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 sum_measure_preimage_singleton (s : Finset β) {f : α → β}
(hf : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y})) : (∑ b ∈ s, μ (f ⁻¹' {b})) = μ (f ⁻¹' ↑s) := by
simp only [← measure_biUnion_finset (pairwiseDisjoint_fiber f s) hf,
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 209 | 211 |
/-
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 | 2,705 | 2,710 | |
/-
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.Topology.UniformSpace.Basic
import Mathlib.Topology.Compactness.Compact
/-!
# Compact sets in uniform spaces
* `compa... | Mathlib/Topology/UniformSpace/Compact.lean | 196 | 210 | |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
/-!
# Polynomia... | theorem lt_sin_mul {x : ℝ} (hx : 0 < x) (hx' : x < 1) : x < sin (π / 2 * x) := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Bounds.lean | 58 | 58 |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
/-!
# Leading terms of Witt vector multiplication
Th... | apply pow_ne_zero
exact mod_cast hp.out.ne_zero
theorem polyOfInterest_vars (n : ℕ) : (polyOfInterest p n).vars ⊆ univ ×ˢ range (n + 1) := by
rw [polyOfInterest_vars_eq]; apply mul_polyOfInterest_vars
theorem peval_polyOfInterest (n : ℕ) (x y : 𝕎 k) :
| Mathlib/RingTheory/WittVector/MulCoeff.lean | 196 | 202 |
/-
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.ComplexShape
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Tactic.ByContra
/-! # Embeddings of comp... | instance [e.IsTruncGE] : e.op.IsTruncLE where
mem_prev h := e.mem_next h
instance [e.IsTruncLE] : e.op.IsTruncGE where
mem_next h := e.mem_prev h
open Classical in
/-- The map `ι' → Option ι` which sends `e.f i` to `some i` and the other elements to `none`. -/
| Mathlib/Algebra/Homology/Embedding/Basic.lean | 130 | 137 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.LinearAlgebra.BilinearMap
/-!
# Images of pairs of submodules under bilinear maps
This file provides `Subm... | on_goal 1 =>
intro a ha
apply @span_induction R N _ _ _ t
· intro b hb
exact subset_span ⟨_, ‹_›, _, ‹_›, rfl⟩
all_goals
intros
simp only [*, add_mem, smul_mem, zero_mem, map_zero, map_add,
LinearMap.zero_apply, LinearMap.add_apply, LinearMap.smul_apply, map_smul]
... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 59 | 73 |
/-
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... | exact one_le_hausdorffMeasure_zero_of_nonempty ⟨f x, hx⟩
· haveI := noAtoms_hausdorff X h'd
simp only [zero_le, measure_singleton]
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 774 | 776 |
/-
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, Patrick Massot
-/
import Mathlib.Topology.Neighborhoods
/-!
# Neighborhoods of a set
In this file we define the filter `𝓝ˢ s` or `nhdsSet s` consisting of all ne... | theorem nhdsSet_union (s t : Set X) : 𝓝ˢ (s ∪ t) = 𝓝ˢ s ⊔ 𝓝ˢ t := by
| Mathlib/Topology/NhdsSet.lean | 135 | 135 |
/-
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... |
theorem stronglyMeasurable_in_set {m : MeasurableSpace α} [TopologicalSpace β] [Zero β] {s : Set α}
{f : α → β} (hs : MeasurableSet s) (hf : StronglyMeasurable f)
(hf_zero : ∀ x, x ∉ s → f x = 0) :
∃ fs : ℕ → α →ₛ β,
| Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 920 | 924 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.V... | UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ := by
let A : MeasurableSpace α := m0
have hmeas : ∀ n, ∀ C, MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} := fun n C =>
measurableSet_le measurable_const (stronglyMeasurable_condExp.mono (hℱ n)).measurable.nnnorm
have hg : MemLp g 1 μ := memLp_one_iff_integrable.2 hint... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 184 | 223 |
/-
Copyright (c) 2019 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.Shapes.IsTerminal
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Initial and terminal objects in a category.
##... | { pt := ⊥_ C
ι := { app := fun _ => initial.to _ } }
| Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 213 | 214 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... |
theorem natDegree_not_mem_eraseLead_support : f.natDegree ∉ (eraseLead f).support := fun h =>
ne_natDegree_of_mem_eraseLead_support h rfl
| Mathlib/Algebra/Polynomial/EraseLead.lean | 89 | 92 |
/-
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... |
theorem HasFPowerSeriesWithinAt.add
(hf : HasFPowerSeriesWithinAt f pf s x) (hg : HasFPowerSeriesWithinAt g pg s x) :
HasFPowerSeriesWithinAt (f + g) (pf + pg) s x := by
rcases (hf.eventually.and hg.eventually).exists with ⟨r, hr⟩
exact ⟨r, hr.1.add hr.2⟩
| Mathlib/Analysis/Analytic/Constructions.lean | 83 | 89 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,025 | 1,028 | |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... |
lemma ωScottContinuous.comp (hg : ωScottContinuous g) (hf : ωScottContinuous f) :
ωScottContinuous (g.comp f) :=
ωScottContinuous.of_monotone_map_ωSup
⟨hg.monotone.comp hf.monotone, by simp [hf.map_ωSup, hg.map_ωSup, map_comp]⟩
| Mathlib/Order/OmegaCompletePartialOrder.lean | 288 | 292 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Basic
import Mathlib.Algebra.Order.GroupWithZero.Unbundled.Defs
import Mathlib.Tactic.Linter.Deprecate... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 526 | 532 | |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | theorem intervalIntegrable_id : IntervalIntegrable (fun x => x) μ a b :=
continuous_id.intervalIntegrable a b
theorem intervalIntegrable_const : IntervalIntegrable (fun _ => c) μ a b :=
continuous_const.intervalIntegrable a b
theorem intervalIntegrable_one_div (h : ∀ x : ℝ, x ∈ [[a, b]] → f x ≠ 0)
(hf : Conti... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 203 | 211 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Ba... | let r := r.toReal
unfold poissonPMFReal
apply (hasSum_mul_left_iff (exp_ne_zero r)).mp
simp only [mul_one]
have : (fun i ↦ rexp r * (rexp (-r) * r ^ i / ↑(Nat.factorial i))) =
fun i ↦ r ^ i / ↑(Nat.factorial i) := by
ext n
rw [mul_div_assoc, exp_neg, ← mul_assoc, ← div_eq_mul_inv, div_self (exp_... | Mathlib/Probability/Distributions/Poisson.lean | 37 | 47 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | conv_lhs => rw [(derivative^[k] p).as_sum_support_C_mul_X_pow]
refine sum_congr rfl fun i _ ↦ ?_
rw [coeff_iterate_derivative, Nat.descFactorial_eq_factorial_mul_choose]
| Mathlib/Algebra/Polynomial/Derivative.lean | 341 | 343 |
/-
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.Data.NNRat.Order
import Mathlib.Topology.Algebra.Order.Archimedean
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.In... | namespace Rat
| Mathlib/Topology/Instances/Rat.lean | 70 | 71 |
/-
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... | section reflGen
lemma reflGen_eq_self (hr : Reflexive r) : ReflGen r = r := by
ext x y
simpa only [reflGen_iff, or_iff_right_iff_imp] using fun h ↦ h ▸ hr y
| Mathlib/Logic/Relation.lean | 445 | 449 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Data.Complex.Trigonometric
/-!
# Trigonometric functions as sums of infinite series
In t... | (Complex.sin z) := by
rw [Complex.sin, Complex.exp_eq_exp_ℂ]
have := (((expSeries_div_hasSum_exp ℂ (-z * Complex.I)).sub
(expSeries_div_hasSum_exp ℂ (z * Complex.I))).mul_right Complex.I).div_const 2
replace := (Nat.divModEquiv 2).symm.hasSum_iff.mpr this
dsimp [Function.comp_def] at this
simp_rw [←... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 49 | 64 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Frédéric Dupuis
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.Equiv.Defs
import Mathlib.Algebra.Module.LinearMa... | variable (A) in
protected lemma IsSelfAdjoint.algebraMap {r : R} (hr : IsSelfAdjoint r) :
IsSelfAdjoint (algebraMap R A r) := by
| Mathlib/Algebra/Star/Module.lean | 217 | 219 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | subset_antisymm (by
simpa using ((closedConvexHull 𝕜).monotone (closure_subset_closedConvexHull (𝕜 := 𝕜) (E := E))))
((closedConvexHull 𝕜).monotone subset_closure)
end TopologicalSpace
section ContinuousConstSMul
| Mathlib/Analysis/Convex/Topology.lean | 349 | 355 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Floris van Doorn
-/
import Mathlib.Topology.VectorBundle.Basic
/-!
# The vector bundle of continuous (semi)linear maps
We define the (topological) vector bundle of ... | end Pretrivialization
open Pretrivialization
variable (F₁ E₁ F₂ E₂)
variable [∀ x : B, TopologicalSpace (E₁ x)] [FiberBundle F₁ E₁] [VectorBundle 𝕜₁ F₁ E₁]
variable [∀ x : B, TopologicalSpace (E₂ x)] [FiberBundle F₂ E₂] [VectorBundle 𝕜₂ F₂ E₂]
| Mathlib/Topology/VectorBundle/Hom.lean | 177 | 183 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | by_cases hx : x = 0
· simp only [hx, zero_dotProduct, star_zero, RCLike.zero_re']
exact le_rfl
· exact le_of_lt (hM.2 x hx)
theorem transpose {M : Matrix n n R} (hM : M.PosDef) : Mᵀ.PosDef := by
| Mathlib/LinearAlgebra/Matrix/PosDef.lean | 348 | 353 |
/-
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
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... |
@[simp]
theorem integrableOn_add_measure :
IntegrableOn f s (μ + ν) ↔ IntegrableOn f s μ ∧ IntegrableOn f s ν :=
⟨fun h =>
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 206 | 210 |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | homEquiv_naturality_left_symm := fun f g => tensorRightHomEquiv_symm_naturality f g
homEquiv_naturality_right := fun f g => tensorRightHomEquiv_naturality f g }
/--
If `Y` has a left dual `ᘁY`, then it is a closed object, with the internal hom functor `Y ⟶[C] -`
| Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 352 | 356 |
/-
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.Algebra.Order.Group.Finset
import Mathlib.Data.Finsupp.Order
import Mathlib.Data.Sym.Basic
/-!
# Equivalence between `Multiset` and `ℕ`-valued finitel... | theorem card_toMultiset (f : α →₀ ℕ) : Multiset.card (toMultiset f) = f.sum fun _ => id := by
simp [toMultiset_apply, map_finsuppSum, Function.id_def]
| Mathlib/Data/Finsupp/Multiset.lean | 61 | 63 |
/-
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.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | pullbackSymmetry_hom_comp_fst]
@[simp, reassoc]
theorem t_fst_snd (i j : 𝒰.J) :
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 64 | 67 |
/-
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.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# Tensor algebras as direct sums of tensor powers
In this file we show... | @[simp]
theorem toDirectSum_ofDirectSum (x : ⨁ n, ⨂[R]^n M) :
TensorAlgebra.toDirectSum (ofDirectSum x) = x :=
AlgHom.congr_fun toDirectSum_comp_ofDirectSum x
/-- The tensor algebra is isomorphic to a direct sum of tensor powers. -/
@[simps!]
def equivDirectSum : TensorAlgebra R M ≃ₐ[R] ⨁ n, ⨂[R]^n M :=
| Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 155 | 162 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | cases hf (Set T) with | intro U hU =>
let g : Set T → T := fun s ↦ ⟨U, cast hU.symm s⟩
have hg : Injective g := by
intro s t h
suffices cast hU (g s).2 = cast hU (g t).2 by
simp only [g, cast_cast, cast_eq] at this
assumption
· congr
exact cantor_injective g hg
| Mathlib/Logic/Function/Basic.lean | 258 | 266 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Eric Wieser
-/
import Mathlib.Data.ENNReal.Holder
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
import Mathl... | (eLpNorm_le_eLpNorm_mul_rpow_measure_univ hpq hf).trans (le_of_eq (by simp [measure_univ]))
theorem eLpNorm'_lt_top_of_eLpNorm'_lt_top_of_exponent_le {p q : ℝ} [IsFiniteMeasure μ]
| Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 100 | 102 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... |
theorem iUnion_image_preimage_sigma_mk_eq_self {ι : Type*} {σ : ι → Type*} (s : Set (Sigma σ)) :
| Mathlib/Data/Set/Lattice.lean | 967 | 968 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | have t_ih_l' := t_ih_l h.left
have t_ih_r' := t_ih_r h.right
| Mathlib/Data/Ordmap/Ordset.lean | 547 | 548 |
/-
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.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... |
/-- A centroid, over the image of an embedding, equals a centroid with
the same points and weights over the original `Finset`. -/
theorem centroid_map (e : ι₂ ↪ ι) (p : ι → P) :
(s₂.map e).centroid k p = s₂.centroid k (p ∘ e) := by
simp [centroid_def, affineCombination_map, centroidWeights]
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 804 | 809 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data... | ext (i x) : 2
induction i with
| fz => rfl
| fs _ ih =>
apply ih
simp only [lastFun_from_append1_drop_last, lastFun_toSubtype, lastFun_appendFun,
lastFun_subtypeVal, Function.id_comp, lastFun_comp, lastFun_prod]
ext1
| Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 378 | 385 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 404 | 406 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Adjoin.FG
/-!
# Adjoining elements and being finitely generated in an algebra tower
## Main results
* `Algebra.fg_trans'`: if `S` is finitely gene... | (hS : Algebra.adjoin C S = ⊤) (hT : Algebra.adjoin C T = ⊤) :
(Algebra.adjoin E (algebraMap D F '' S)).restrictScalars C =
(Algebra.adjoin D (algebraMap E F '' T)).restrictScalars C := by
rw [adjoin_restrictScalars C E, adjoin_restrictScalars C D, ← hS, ← hT, ← Algebra.adjoin_image,
← Algebra.adjoin... | Mathlib/RingTheory/Adjoin/Tower.lean | 49 | 58 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | let ⟨_, vc, pc⟩ ← evalMul sα vb vb
return ⟨_, vc, (q(pow_bit0 $pb $pc) : Expr)⟩
| Mathlib/Tactic/Ring/Basic.lean | 748 | 749 |
/-
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 | 1,452 | 1,454 | |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Data.Set.UnionLift
import Mathlib.LinearAlgebra.Span.Basic
import Mathlib.RingTheory.NonUnitalSubring.Basic
... | { NonUnitalSubsemiring.center A with smul_mem' := Set.smul_mem_center }
theorem coe_center : (center R A : Set A) = Set.center A :=
| Mathlib/Algebra/Algebra/NonUnitalSubalgebra.lean | 1,072 | 1,074 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
open Submodule
namespace Module
| Mathlib/Algebra/Module/Torsion.lean | 296 | 300 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... | theorem coe_zero : (↑(0 : ℚ) : ℚ_[p]) = 0 := rfl
end Completion
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 519 | 521 |
/-
Copyright (c) 2022 Yaël Dillies, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Kexing Ying
-/
import Mathlib.Analysis.Normed.Order.UpperLower
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.Topology.Order.Dense... |
Along with `aux₀`, this proves that `x` is a Lebesgue point of `s`. This will be used to prove that
the frontier of an order-connected set is null. -/
private lemma aux₁
(h : ∀ δ, 0 < δ →
∃ y, closedBall y (δ / 4) ⊆ closedBall x δ ∧ closedBall y (δ / 4) ⊆ interior sᶜ) :
¬Tendsto (fun r ↦ volume (closure ... | Mathlib/MeasureTheory/Order/UpperLower.lean | 84 | 123 |
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 1,384 | 1,386 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.LinearAlgebra.TensorProduct.Graded.Internal
import Mathlib.LinearAlgebra.QuadraticForm.Prod
/-!
# Cliff... |
theorem map_mul_map_eq_neg_of_isOrtho_of_mem_evenOdd_one
(hm₁ : m₁ ∈ evenOdd Q₁ 1) (hm₂ : m₂ ∈ evenOdd Q₂ 1) :
map f₁ m₁ * map f₂ m₂ = - map f₂ m₂ * map f₁ m₁ := by
| Mathlib/LinearAlgebra/CliffordAlgebra/Prod.lean | 101 | 104 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Measure.Decomposition.Exhaustion
import Mathlib.MeasureTheory.Me... | simp only [Pi.smul_apply, smul_eq_mul]
theorem withDensity_smul' (r : ℝ≥0∞) (f : α → ℝ≥0∞) (hr : r ≠ ∞) :
μ.withDensity (r • f) = r • μ.withDensity f := by
refine Measure.ext fun s hs => ?_
rw [withDensity_apply _ hs, Measure.coe_smul, Pi.smul_apply, withDensity_apply _ hs,
| Mathlib/MeasureTheory/Measure/WithDensity.lean | 122 | 127 |
/-
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
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Matr... | theorem trace_mul_comm [AddCommMonoid R] [CommMagma R] (A : Matrix m n R) (B : Matrix n m R) :
trace (A * B) = trace (B * A) := by rw [← trace_transpose, ← trace_transpose_mul, transpose_mul]
| Mathlib/LinearAlgebra/Matrix/Trace.lean | 154 | 155 |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... | variable {s : Set α} (hs : sSupIndep s)
@[simp]
theorem sSupIndep_empty : sSupIndep (∅ : Set α) := fun x hx =>
(Set.not_mem_empty x hx).elim
@[deprecated (since := "2024-11-24")] alias CompleteLattice.setIndependent_empty := sSupIndep_empty
include hs in
theorem sSupIndep.mono {t : Set α} (hst : t ⊆ s) : sSupIndep... | Mathlib/Order/SupIndep.lean | 246 | 261 |
/-
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... |
theorem SameCycle.of_pow {n : ℕ} : SameCycle (f ^ n) x y → SameCycle f x y := fun ⟨m, h⟩ =>
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 157 | 158 |
/-
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.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | theorem one_mem_properDivisors_iff_one_lt : 1 ∈ n.properDivisors ↔ 1 < n := by
rw [mem_properDivisors, and_iff_right (one_dvd _)]
| Mathlib/NumberTheory/Divisors.lean | 264 | 266 |
/-
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.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | then the image is measurable. -/
theorem measurable_image_of_fderivWithin (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) : MeasurableSet (f '' s) :=
haveI : DifferentiableOn ℝ f s := fun x hx => (hf' x hx).differentiableWithinAt
hs.image_of_continuousOn_injOn (Differenti... | Mathlib/MeasureTheory/Function/Jacobian.lean | 760 | 766 |
/-
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... | toIocMod hp a (b + c) = toIocMod hp (a - c) b + c := by
simp_rw [toIocMod, toIocDiv_sub_eq_toIocDiv_add', sub_add_eq_add_sub]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 453 | 454 |
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 838 | 842 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | simp only [UniformContinuous, uniformity_prod_eq_prod, uniformity_eq, prod_map_map_eq,
tendsto_map'_iff]
rfl
theorem uniformContinuous_lift {f : α → β} (h : ∀ a b, Inseparable a b → f a = f b) :
UniformContinuous (lift f h) ↔ UniformContinuous f :=
| Mathlib/Topology/UniformSpace/Separation.lean | 234 | 239 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | (cons g u w h1 h2).prod φ = of g * (t ^ (u : ℤ) * w.prod φ) := by
simp [ReducedWord.prod, cons, smul_def, mul_assoc]
theorem prod_unitsSMul (u : ℤˣ) (w : NormalWord d) :
(unitsSMul φ u w).prod φ = (t^(u : ℤ) * w.prod φ : HNNExtension G A B φ) := by
| Mathlib/GroupTheory/HNNExtension.lean | 508 | 512 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | theorem rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n⁻¹ : ℝ)) ^ n = x := by
have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 488 | 489 |
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib... | · intro h
simpa only [inv_eq_inv, IsIso.inv_inv] using S.inv h
· apply S.inv
theorem mul_mem_cancel_left {c d e : C} {f : c ⟶ d} {g : d ⟶ e} (hf : f ∈ S.arrows c d) :
f ≫ g ∈ S.arrows c e ↔ g ∈ S.arrows d e := by
| Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 82 | 87 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Justus Springer
-/
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
/-!
# Stalks
For a presheaf `F` on a topol... | -- Since `F` is a sheaf, we can prove the equality locally, if we can show that these
-- neighborhoods form a cover of `U`.
apply F.eq_of_locally_eq' V U i₁
· intro x hxU
simp only [Opens.coe_iSup, Set.mem_iUnion, SetLike.mem_coe]
exact ⟨⟨x, hxU⟩, m ⟨x, hxU⟩⟩
· intro x
rw [heq, Subsingleton.elim (... | Mathlib/Topology/Sheaves/Stalks.lean | 449 | 461 |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Analysis.SpecialFunctions.Complex.LogBounds
/-!
# Complex arctangent
This file defines the complex arctangent `Complex.arctan` as
$$\arctan z = -\frac i2 \lo... | /-- We can combine the logs in `log (1 + z * I) + -log (1 - z * I)` into one.
This is only used in `hasSum_arctan`. -/
lemma hasSum_arctan_aux {z : ℂ} (hz : ‖z‖ < 1) :
log (1 + z * I) + -log (1 - z * I) = log ((1 + z * I) / (1 - z * I)) := by
have z₁ := mem_slitPlane_iff_arg.mp (mem_slitPlane_of_norm_lt_one (z :=... | Mathlib/Analysis/SpecialFunctions/Complex/Arctan.lean | 93 | 112 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
variable [Preorder α] [PartialOrder β] [LocallyFiniteOrder α] [LocallyFiniteOrder β]
[DecidableLE (α × β)] (a b : α) (c : β)
| Mathlib/Order/Interval/Finset/Basic.lean | 675 | 677 |
/-
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 ... | rwa [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]
have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by
rw [mem_preimage]
exact hxs
have := this.subset_connectedComponent h2xs
rw [connectedComponentIn_eq_image (hsF hxs)]
refine Subset.trans ?_ (image_subset _ this)
rw [Subtype.image_preimage_coe... | Mathlib/Topology/Connected/Basic.lean | 523 | 531 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | rw [← div_spanSingleton, le_div_iff_mul_le h_spanx] at hJ
obtain ⟨s, hs⟩ := hI _ hJ
use s * {(algebraMap R₁ K x)⁻¹}
rw [Finset.coe_mul, Finset.coe_singleton, ← span_mul_span, hs, ← coe_spanSingleton R₁⁰, ←
coe_mul, mul_assoc, spanSingleton_mul_spanSingleton, mul_inv_cancel₀ h_gx, spanSingleton_one,
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 854 | 858 |
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