Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e... | Mathlib/CategoryTheory/Subobject/Limits.lean | 309 | 310 | theorem imageSubobject_arrow :
(imageSubobjectIso f).hom ≫ image.ι f = (imageSubobject f).arrow := by | simp [imageSubobjectIso]
|
/-
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.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 742 | 757 | theorem orthonormal_iff_ite [DecidableEq ι] {v : ι → E} :
Orthonormal 𝕜 v ↔ ∀ i j, ⟪v i, v j⟫ = if i = j then (1 : 𝕜) else (0 : 𝕜) := by |
constructor
· intro hv i j
split_ifs with h
· simp [h, inner_self_eq_norm_sq_to_K, hv.1]
· exact hv.2 h
· intro h
constructor
· intro i
have h' : ‖v i‖ ^ 2 = 1 ^ 2 := by simp [@norm_sq_eq_inner 𝕜, h i i]
have h₁ : 0 ≤ ‖v i‖ := norm_nonneg _
have h₂ : (0 : ℝ) ≤ 1 := zero_le_... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.Preserves.Limits
#align_import category_theory.limits.functor_category from "lean... | Mathlib/CategoryTheory/Limits/FunctorCategory.lean | 235 | 239 | theorem limitObjIsoLimitCompEvaluation_inv_limit_map [HasLimitsOfShape J C] {i j : K}
(F : J ⥤ K ⥤ C) (f : i ⟶ j) : (limitObjIsoLimitCompEvaluation _ _).inv ≫ (limit F).map f =
limMap (whiskerLeft _ ((evaluation _ _).map f)) ≫ (limitObjIsoLimitCompEvaluation _ _).inv := by |
rw [Iso.inv_comp_eq, ← Category.assoc, Iso.eq_comp_inv,
limit_map_limitObjIsoLimitCompEvaluation_hom]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 112 | 115 | theorem exists_rat_eq_nth_denominator : ∃ q : ℚ, (of v).denominators n = (q : K) := by |
rcases exists_gcf_pair_rat_eq_nth_conts v n with ⟨⟨_, b⟩, nth_cont_eq⟩
use b
simp [denom_eq_conts_b, nth_cont_eq]
|
/-
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.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingThe... | Mathlib/RingTheory/Unramified/Basic.lean | 222 | 227 | theorem localization_map [FormallyUnramified R S] :
FormallyUnramified Rₘ Sₘ := by |
haveI : FormallyUnramified S Sₘ :=
FormallyUnramified.of_isLocalization (M.map (algebraMap R S))
haveI : FormallyUnramified R Sₘ := FormallyUnramified.comp R S Sₘ
exact FormallyUnramified.localization_base M
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
#align_import model_theory.substructures from "lea... | Mathlib/ModelTheory/Substructures.lean | 335 | 339 | theorem _root_.Set.Countable.substructure_closure
[Countable (Σl, L.Functions l)] (h : s.Countable) : Countable.{w + 1} (closure L s) := by |
haveI : Countable s := h.to_subtype
rw [← mk_le_aleph0_iff, ← lift_le_aleph0]
exact lift_card_closure_le_card_term.trans mk_le_aleph0
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
i... | Mathlib/Analysis/Calculus/Rademacher.lean | 97 | 117 | theorem integral_inv_smul_sub_mul_tendsto_integral_lineDeriv_mul
(hf : LipschitzWith C f) (hg : Integrable g μ) (v : E) :
Tendsto (fun (t : ℝ) ↦ ∫ x, (t⁻¹ • (f (x + t • v) - f x)) * g x ∂μ) (𝓝[>] 0)
(𝓝 (∫ x, lineDeriv ℝ f x v * g x ∂μ)) := by |
apply tendsto_integral_filter_of_dominated_convergence (fun x ↦ (C * ‖v‖) * ‖g x‖)
· filter_upwards with t
apply AEStronglyMeasurable.mul ?_ hg.aestronglyMeasurable
apply aestronglyMeasurable_const.smul
apply AEStronglyMeasurable.sub _ hf.continuous.measurable.aestronglyMeasurable
apply AEMeasurabl... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 263 | 272 | theorem pullback_snd_range {X Y S : TopCat} (f : X ⟶ S) (g : Y ⟶ S) :
Set.range (pullback.snd : pullback f g ⟶ _) = { y : Y | ∃ x : X, f x = g y } := by |
ext y
constructor
· rintro ⟨(x : (forget TopCat).obj _), rfl⟩
use (pullback.fst : pullback f g ⟶ _) x
exact ConcreteCategory.congr_hom pullback.condition x
· rintro ⟨x, eq⟩
use (TopCat.pullbackIsoProdSubtype f g).inv ⟨⟨x, y⟩, eq⟩
rw [pullbackIsoProdSubtype_inv_snd_apply]
|
/-
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, Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Ma... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 260 | 271 | theorem Submodule.exists_le_ker_of_lt_top (p : Submodule K V) (hp : p < ⊤) :
∃ (f : V →ₗ[K] K), f ≠ 0 ∧ p ≤ ker f := by |
rcases SetLike.exists_of_lt hp with ⟨v, -, hpv⟩; clear hp
rcases (LinearPMap.supSpanSingleton ⟨p, 0⟩ v (1 : K) hpv).toFun.exists_extend with ⟨f, hf⟩
refine ⟨f, ?_, ?_⟩
· rintro rfl
rw [LinearMap.zero_comp] at hf
have := LinearPMap.supSpanSingleton_apply_mk ⟨p, 0⟩ v (1 : K) hpv 0 p.zero_mem 1
simpa ... |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 609 | 611 | theorem nmul_le_nmul_of_nonneg_right (h₁ : a ≤ b) (h₂ : 0 ≤ c) : a ⨳ c ≤ b ⨳ c := by |
rw [nmul_comm, nmul_comm b]
exact nmul_le_nmul_of_nonneg_left h₁ h₂
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 106 | 122 | theorem uniformEquicontinuous_birkhoffAverage (hf : LipschitzWith 1 f) (hg : UniformContinuous g) :
UniformEquicontinuous (birkhoffAverage 𝕜 f g) := by |
refine Metric.uniformity_basis_dist_le.uniformEquicontinuous_iff_right.2 fun ε hε ↦ ?_
rcases (uniformity_basis_edist_le.uniformContinuous_iff Metric.uniformity_basis_dist_le).1 hg ε hε
with ⟨δ, hδ₀, hδε⟩
refine mem_uniformity_edist.2 ⟨δ, hδ₀, fun {x y} h n ↦ ?_⟩
calc
dist (birkhoffAverage 𝕜 f g n x) ... |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 430 | 431 | theorem cthickening_closure : cthickening δ (closure s) = cthickening δ s := by |
simp_rw [cthickening, infEdist_closure]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 54 | 54 | theorem eq_false_eq_not_eq_true (b : Bool) : (¬b = true) = (b = false) := by | simp
|
/-
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.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanpro... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 278 | 288 | theorem subset_positive_null_set (hu : MeasurableSet u) (hv : MeasurableSet v)
(hw : MeasurableSet w) (hsu : 0 ≤[u] s) (hw₁ : s w = 0) (hw₂ : w ⊆ u) (hwt : v ⊆ w) :
s v = 0 := by |
have : s v + s (w \ v) = 0 := by
rw [← hw₁, ← of_union Set.disjoint_sdiff_right hv (hw.diff hv), Set.union_diff_self,
Set.union_eq_self_of_subset_left hwt]
have h₁ := nonneg_of_zero_le_restrict _ (restrict_le_restrict_subset _ _ hu hsu (hwt.trans hw₂))
have h₂ : 0 ≤ s (w \ v) :=
nonneg_of_zero_le_r... |
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 185 | 192 | theorem approx_le_one (c : CU P) (n : ℕ) (x : X) : c.approx n x ≤ 1 := by |
induction' n with n ihn generalizing c
· exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one
· simp only [approx, midpoint_eq_smul_add, invOf_eq_inv, smul_eq_mul, ← div_eq_inv_mul]
have := add_le_add (ihn (left c)) (ihn (right c))
set_option tactic.skipAssignedInstances false in
norm_num... |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 121 | 122 | theorem goldConj_neg : ψ < 0 := by |
linarith [one_sub_goldConj, one_lt_gold]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 696 | 698 | theorem integral_add (f g : α →₁[μ] E) : integral (f + g) = integral f + integral g := by |
simp only [integral]
exact map_add integralCLM f g
|
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 305 | 309 | theorem mem_factors_mul_left {p a b : ℕ} (hpa : p ∈ a.factors) (hb : b ≠ 0) :
p ∈ (a * b).factors := by |
rcases eq_or_ne a 0 with (rfl | ha)
· simp at hpa
apply (mem_factors_mul ha hb).2 (Or.inl hpa)
|
/-
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.... | Mathlib/ModelTheory/Semantics.lean | 376 | 392 | theorem realize_mapTermRel_add_castLe [L'.Structure M] {k : ℕ}
{ft : ∀ n, L.Term (Sum α (Fin n)) → L'.Term (Sum β (Fin (k + n)))}
{fr : ∀ n, L.Relations n → L'.Relations n} {n} {φ : L.BoundedFormula α n}
(v : ∀ {n}, (Fin (k + n) → M) → α → M) {v' : β → M} (xs : Fin (k + n) → M)
(h1 :
∀ (n) (t : L.... |
induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih
· rfl
· simp [mapTermRel, Realize, h1]
· simp [mapTermRel, Realize, h1, h2]
· simp [mapTermRel, Realize, ih1, ih2]
· simp [mapTermRel, Realize, ih, hv]
|
/-
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.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.b... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 278 | 282 | theorem Basis.toMatrix_reindex (b : Basis ι R M) (v : ι' → M) (e : ι ≃ ι') :
(b.reindex e).toMatrix v = (b.toMatrix v).submatrix e.symm _root_.id := by |
ext
simp only [Basis.toMatrix_apply, Basis.repr_reindex, Matrix.submatrix_apply, _root_.id,
Finsupp.mapDomain_equiv_apply]
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,507 | 1,509 | theorem nextCoeff_X_add_C [Semiring S] (c : S) : nextCoeff (X + C c) = c := by |
nontriviality S
simp [nextCoeff_of_natDegree_pos]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | Mathlib/Topology/Instances/AddCircle.lean | 405 | 414 | theorem addOrderOf_period_div {n : ℕ} (h : 0 < n) : addOrderOf ((p / n : 𝕜) : AddCircle p) = n := by |
rw [addOrderOf_eq_iff h]
replace h : 0 < (n : 𝕜) := Nat.cast_pos.2 h
refine ⟨?_, fun m hn h0 => ?_⟩ <;> simp only [Ne, ← coe_nsmul, nsmul_eq_mul]
· rw [mul_div_cancel₀ _ h.ne', coe_period]
rw [coe_eq_zero_of_pos_iff p hp.out (mul_pos (Nat.cast_pos.2 h0) <| div_pos hp.out h)]
rintro ⟨k, hk⟩
rw [mul_div, ... |
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 1,281 | 1,287 | theorem coe_mkFinCons {n : ℕ} {N : Submodule R M} (y : M) (b : Basis (Fin n) R N)
(hli : ∀ (c : R), ∀ x ∈ N, c • y + x = 0 → c = 0) (hsp : ∀ z : M, ∃ c : R, z + c • y ∈ N) :
(mkFinCons y b hli hsp : Fin (n + 1) → M) = Fin.cons y ((↑) ∘ b) := by |
-- Porting note: without `unfold`, Lean can't reuse the proofs included in the definition
-- `mkFinCons`
unfold mkFinCons
exact coe_mk (v := Fin.cons y (N.subtype ∘ b)) _ _
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 647 | 648 | theorem map_inv_hom (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [IsIso f] :
F.map (inv f) ≫ F.map f = 𝟙 (F.obj Y) := by | simp
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... | Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 146 | 151 | theorem map_ext' {f g : AdicCompletion I M →ₗ[AdicCompletion I R] T}
(h : ∀ (a : AdicCauchySequence I M),
f (AdicCompletion.mk I M a) = g (AdicCompletion.mk I M a)) :
f = g := by |
ext x
apply induction_on I M x (fun a ↦ h a)
|
/-
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.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 33 | 37 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by |
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
|
/-
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.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,216 | 1,217 | theorem sup_iSup [Nonempty ι] {f : ι → α} {a : α} : (a ⊔ ⨆ x, f x) = ⨆ x, a ⊔ f x := by |
rw [iSup_sup_eq, iSup_const]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 237 | 242 | theorem ENNReal.tendsto_pow_atTop_nhds_zero_of_lt_one {r : ℝ≥0∞} (hr : r < 1) :
Tendsto (fun n : ℕ ↦ r ^ n) atTop (𝓝 0) := by |
rcases ENNReal.lt_iff_exists_coe.1 hr with ⟨r, rfl, hr'⟩
rw [← ENNReal.coe_zero]
norm_cast at *
apply NNReal.tendsto_pow_atTop_nhds_zero_of_lt_one hr
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 648 | 666 | theorem _root_.PartialHomeomorph.isLocalStructomorphWithinAt_iff {G : StructureGroupoid H}
[ClosedUnderRestriction G] (f : PartialHomeomorph H H) {s : Set H} {x : H}
(hx : x ∈ f.source ∪ sᶜ) :
G.IsLocalStructomorphWithinAt (⇑f) s x ↔
x ∈ s → ∃ e : PartialHomeomorph H H,
e ∈ G ∧ e.source ⊆ f.sour... |
constructor
· intro hf h2x
obtain ⟨e, he, hfe, hxe⟩ := hf h2x
refine ⟨e.restr f.source, closedUnderRestriction' he f.open_source, ?_, ?_, hxe, ?_⟩
· simp_rw [PartialHomeomorph.restr_source]
exact inter_subset_right.trans interior_subset
· intro x' hx'
exact hfe ⟨hx'.1, hx'.2.1⟩
· rw... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.... | Mathlib/RingTheory/Localization/Submodule.lean | 75 | 78 | theorem coeSubmodule_span (s : Set R) :
coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by |
rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]
rfl
|
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
impor... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 927 | 928 | theorem lintegral_coeFn (f : α →ₘ[μ] ℝ≥0∞) : ∫⁻ a, f a ∂μ = f.lintegral := by |
rw [← lintegral_mk, mk_coeFn]
|
/-
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.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 594 | 597 | theorem Prod.tendsto_iff {X} (seq : X → Y × Z) {f : Filter X} (p : Y × Z) :
Tendsto seq f (𝓝 p) ↔
Tendsto (fun n => (seq n).fst) f (𝓝 p.fst) ∧ Tendsto (fun n => (seq n).snd) f (𝓝 p.snd) := by |
rw [nhds_prod_eq, Filter.tendsto_prod_iff']
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.PUnit
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib... | Mathlib/CategoryTheory/Comma/Over.lean | 81 | 81 | theorem w {A B : Over X} (f : A ⟶ B) : f.left ≫ B.hom = A.hom := by | have := f.w; aesop_cat
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 87 | 90 | theorem Path.cast_eq_cast {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Path u v) :
p.cast hu hv = _root_.cast (by rw [hu, hv]) p := by |
subst_vars
rfl
|
/-
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.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCa... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 130 | 161 | theorem exists_locallyConstant_finite_aux {α : Type*} [Finite α] (hC : IsLimit C)
(f : LocallyConstant C.pt α) : ∃ (j : J) (g : LocallyConstant (F.obj j) (α → Fin 2)),
(f.map fun a b => if a = b then (0 : Fin 2) else 1) = g.comap (C.π.app _) := by |
cases nonempty_fintype α
let ι : α → α → Fin 2 := fun x y => if x = y then 0 else 1
let ff := (f.map ι).flip
have hff := fun a : α => exists_locallyConstant_fin_two _ hC (ff a)
choose j g h using hff
let G : Finset J := Finset.univ.image j
obtain ⟨j0, hj0⟩ := IsCofiltered.inf_objs_exists G
have hj : ∀ ... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 154 | 167 | theorem integral_inner_eq_sq_snorm (f : α →₂[μ] E) :
∫ a, ⟪f a, f a⟫ ∂μ = ENNReal.toReal (∫⁻ a, (‖f a‖₊ : ℝ≥0∞) ^ (2 : ℝ) ∂μ) := by |
simp_rw [inner_self_eq_norm_sq_to_K]
norm_cast
rw [integral_eq_lintegral_of_nonneg_ae]
rotate_left
· exact Filter.eventually_of_forall fun x => sq_nonneg _
· exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable
congr
ext1 x
have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by ... |
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.specia... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 224 | 232 | theorem stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq (n : ℕ) (hn : n ≠ 0) :
stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (n / (2 * n + 1)) = Wallis.W n := by |
have : 4 = 2 * 2 := by rfl
rw [stirlingSeq, this, pow_mul, stirlingSeq, Wallis.W_eq_factorial_ratio]
simp_rw [div_pow, mul_pow]
rw [sq_sqrt, sq_sqrt]
any_goals positivity
field_simp [← exp_nsmul]
ring_nf
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.cartan_subalgebra from "leanprover-community/mathlib"@"938fead7abdc0cb... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 65 | 69 | theorem ucs_eq_self_of_isCartanSubalgebra (H : LieSubalgebra R L) [H.IsCartanSubalgebra] (k : ℕ) :
H.toLieSubmodule.ucs k = H.toLieSubmodule := by |
induction' k with k ih
· simp
· simp [ih]
|
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 593 | 606 | theorem lintegral_sum_measure {m : MeasurableSpace α} {ι} (f : α → ℝ≥0∞) (μ : ι → Measure α) :
∫⁻ a, f a ∂Measure.sum μ = ∑' i, ∫⁻ a, f a ∂μ i := by |
simp only [lintegral, iSup_subtype', SimpleFunc.lintegral_sum, ENNReal.tsum_eq_iSup_sum]
rw [iSup_comm]
congr; funext s
induction' s using Finset.induction_on with i s hi hs
· simp
simp only [Finset.sum_insert hi, ← hs]
refine (ENNReal.iSup_add_iSup ?_).symm
intro φ ψ
exact
⟨⟨φ ⊔ ψ, fun x => sup_... |
/-
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.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 234 | 238 | theorem val_eq_iff_norm_eq {f g : PadicSeq p} (hf : ¬f ≈ 0) (hg : ¬g ≈ 0) :
f.valuation = g.valuation ↔ f.norm = g.norm := by |
rw [norm_eq_pow_val hf, norm_eq_pow_val hg, ← neg_inj, zpow_inj]
· exact mod_cast (Fact.out : p.Prime).pos
· exact mod_cast (Fact.out : p.Prime).ne_one
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 622 | 623 | theorem Sublist.filter (p : α → Bool) {l₁ l₂} (s : l₁ <+ l₂) : filter p l₁ <+ filter p l₂ := by |
rw [← filterMap_eq_filter]; apply s.filterMap
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 3,292 | 3,295 | theorem toFinset_append : toFinset (l ++ l') = l.toFinset ∪ l'.toFinset := by |
induction' l with hd tl hl
· simp
· simp [hl]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Ma... | Mathlib/Analysis/Analytic/Basic.lean | 335 | 342 | theorem min_radius_le_radius_add (p q : FormalMultilinearSeries 𝕜 E F) :
min p.radius q.radius ≤ (p + q).radius := by |
refine ENNReal.le_of_forall_nnreal_lt fun r hr => ?_
rw [lt_min_iff] at hr
have := ((p.isLittleO_one_of_lt_radius hr.1).add (q.isLittleO_one_of_lt_radius hr.2)).isBigO
refine (p + q).le_radius_of_isBigO ((isBigO_of_le _ fun n => ?_).trans this)
rw [← add_mul, norm_mul, norm_mul, norm_norm]
exact mul_le_mul... |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 73 | 82 | theorem countable_meas_le_ne_meas_lt (g : α → R) :
{t : R | μ {a : α | t ≤ g a} ≠ μ {a : α | t < g a}}.Countable := by |
-- the target set is contained in the set of points where the function `t ↦ μ {a : α | t ≤ g a}`
-- jumps down on the right of `t`. This jump set is countable for any function.
let F : R → ℝ≥0∞ := fun t ↦ μ {a : α | t ≤ g a}
apply (countable_image_gt_image_Ioi F).mono
intro t ht
have : μ {a | t < g a} < μ ... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 267 | 275 | theorem toMatrix_sumInl (t : TransvectionStruct n R) :
(t.sumInl p).toMatrix = fromBlocks t.toMatrix 0 0 1 := by |
cases t
ext a b
cases' a with a a <;> cases' b with b b
· by_cases h : a = b <;> simp [TransvectionStruct.sumInl, transvection, h, stdBasisMatrix]
· simp [TransvectionStruct.sumInl, transvection]
· simp [TransvectionStruct.sumInl, transvection]
· by_cases h : a = b <;> simp [TransvectionStruct.sumInl, tr... |
/-
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
import Mathlib.LinearAlgebra.BilinearMap
#align_import algebra.module.submodule.bilinear from "leanprover-community/mathlib"@"60... | Mathlib/Algebra/Module/Submodule/Bilinear.lean | 59 | 73 | theorem map₂_span_span (f : M →ₗ[R] N →ₗ[R] P) (s : Set M) (t : Set N) :
map₂ f (span R s) (span R t) = span R (Set.image2 (fun m n => f m n) s t) := by |
apply le_antisymm
· rw [map₂_le]
apply @span_induction' R M _ _ _ s
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, _root_.map_zero, map_add,
Linear... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 178 | 179 | theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by |
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.Catego... | Mathlib/CategoryTheory/Equivalence.lean | 351 | 354 | theorem invFunIdAssoc_inv_app (e : C ≌ D) (F : D ⥤ E) (X : D) :
(invFunIdAssoc e F).inv.app X = F.map (e.counitInv.app X) := by |
dsimp [invFunIdAssoc]
aesop_cat
|
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Data.Fintype.Perm
import Mathlib.LinearAlgebra.Multilinear.Basis
#align_import linear_algebra.... | Mathlib/LinearAlgebra/Alternating/Basic.lean | 913 | 915 | theorem alternatization_apply (m : MultilinearMap R (fun _ : ι => M) N') (v : ι → M) :
alternatization m v = ∑ σ : Perm ι, Equiv.Perm.sign σ • m.domDomCongr σ v := by |
simp only [alternatization_def, smul_apply, sum_apply]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebr... | Mathlib/Algebra/Ring/BooleanRing.lean | 105 | 105 | theorem mul_one_add_self : a * (1 + a) = 0 := by | rw [mul_add, mul_one, mul_self, add_self]
|
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | Mathlib/Topology/Compactification/OnePoint.lean | 144 | 145 | theorem ne_infty_iff_exists {x : OnePoint X} : x ≠ ∞ ↔ ∃ y : X, (y : OnePoint X) = x := by |
induction x using OnePoint.rec <;> simp
|
/-
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, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,084 | 2,086 | theorem mk_eq_nat_iff_finset {n : ℕ} :
#α = n ↔ ∃ t : Finset α, (t : Set α) = univ ∧ t.card = n := by |
rw [← mk_univ, mk_set_eq_nat_iff_finset]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 308 | 331 | theorem outer_trim : f.outer.trim = f.outer := by |
refine le_antisymm (fun s => ?_) (OuterMeasure.le_trim _)
rw [OuterMeasure.trim_eq_iInf]
refine le_iInf fun t => le_iInf fun ht => ENNReal.le_of_forall_pos_le_add fun ε ε0 h => ?_
rcases ENNReal.exists_pos_sum_of_countable (ENNReal.coe_pos.2 ε0).ne' ℕ with ⟨ε', ε'0, hε⟩
refine le_trans ?_ (add_le_add_left (l... |
/-
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.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 857 | 857 | theorem floor_eq_zero_iff : ⌊a⌋ = 0 ↔ a ∈ Ico (0 : α) 1 := by | simp [floor_eq_iff]
|
/-
Copyright (c) 2023 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.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.inv from "leanpro... | Mathlib/Analysis/Calculus/Deriv/Inv.lean | 195 | 198 | theorem HasDerivAt.div (hc : HasDerivAt c c' x) (hd : HasDerivAt d d' x) (hx : d x ≠ 0) :
HasDerivAt (fun y => c y / d y) ((c' * d x - c x * d') / d x ^ 2) x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.div hd hx
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | Mathlib/Topology/UniformSpace/Ascoli.lean | 327 | 345 | theorem EquicontinuousOn.tendsto_uniformOnFun_iff_pi'
{𝔖 : Set (Set X)} (𝔖_compact : ∀ K ∈ 𝔖, IsCompact K)
(F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) (ℱ : Filter ι) (f : X → α) :
Tendsto (UniformOnFun.ofFun 𝔖 ∘ F) ℱ (𝓝 <| UniformOnFun.ofFun 𝔖 f) ↔
Tendsto ((⋃₀ 𝔖).restrict ∘ F) ℱ (𝓝 <| (⋃₀ 𝔖).r... |
-- Recall that the uniform structure on `X →ᵤ[𝔖] α` is the one induced by all the maps
-- `K.restrict : (X →ᵤ[𝔖] α) → (K →ᵤ α)` for `K ∈ 𝔖`.
-- Similarly, the uniform structure on `X → α` induced by the map
-- `(⋃₀ 𝔖).restrict : (X → α) → ((⋃₀ 𝔖) → α)` is equal to the one induced by
-- all maps `K.restr... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87... | Mathlib/Algebra/Group/Indicator.lean | 363 | 365 | theorem mulIndicator_union_of_not_mem_inter (h : a ∉ s ∩ t) (f : α → M) :
mulIndicator (s ∪ t) f a = mulIndicator s f a * mulIndicator t f a := by |
rw [← mulIndicator_union_mul_inter_apply f s t, mulIndicator_of_not_mem h, mul_one]
|
/-
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.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,673 | 1,684 | theorem linearIndependent_comp_of_eval
(h₁: ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C (ord I · < o))))) :
LinearIndependent ℤ (eval (C' C ho)) →
LinearIndependent ℤ (ModuleCat.ofHom (Linear_CC' C hsC ho) ∘ SumEval C ho ∘ Sum.inr) := by |
dsimp [SumEval, ModuleCat.ofHom]
erw [max_eq_eval_unapply C hsC ho]
intro h
let f := MaxToGood C hC hsC ho h₁
have hf : f.Injective := maxToGood_injective C hC hsC ho h₁
have hh : (fun l ↦ Products.eval (C' C ho) l.val.Tail) = eval (C' C ho) ∘ f := rfl
rw [hh]
exact h.comp f hf
|
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
/-!
# Local graph operations
This file defines some single-graph operations th... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 82 | 86 | theorem edgeSet_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeSet =
(G.edgeSet \ G.incidenceSet t ∪ (s(·, t)) '' (G.neighborSet s)) \ {s(t, t)} := by |
ext e; refine e.inductionOn ?_
simp only [replaceVertex, mem_edgeSet, Set.mem_union, Set.mem_diff, mk'_mem_incidenceSet_iff]
intros; split_ifs; exacts [by simp_all, by aesop, by rw [adj_comm]; aesop, by aesop]
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 583 | 585 | theorem iterate_eq_of_map_eq_add_int {x : ℝ} {m : ℤ} (h : f x = x + m) (n : ℕ) :
f^[n] x = x + n * m := by |
simpa only [nsmul_eq_mul, add_right_iterate] using (f.commute_add_int m).iterate_eq_of_map_eq n h
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpac... | Mathlib/Analysis/Convex/Segment.lean | 152 | 155 | theorem mem_openSegment_of_ne_left_right (hx : x ≠ z) (hy : y ≠ z) (hz : z ∈ [x -[𝕜] y]) :
z ∈ openSegment 𝕜 x y := by |
rw [← insert_endpoints_openSegment] at hz
exact (hz.resolve_left hx.symm).resolve_left hy.symm
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprove... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 180 | 181 | theorem convergents'Aux_succ_none {s : Stream'.Seq (Pair K)} (h : s.head = none) (n : ℕ) :
convergents'Aux s (n + 1) = 0 := by | simp [convergents'Aux, h, convergents'Aux.match_1]
|
/-
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.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 305 | 309 | theorem snorm_const (c : F) (h0 : p ≠ 0) (hμ : μ ≠ 0) :
snorm (fun _ : α => c) p μ = (‖c‖₊ : ℝ≥0∞) * μ Set.univ ^ (1 / ENNReal.toReal p) := by |
by_cases h_top : p = ∞
· simp [h_top, snormEssSup_const c hμ]
simp [snorm_eq_snorm' h0 h_top, snorm'_const, ENNReal.toReal_pos h0 h_top]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymp... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 128 | 130 | theorem isEquivalent_zero_iff_eventually_zero : u ~[l] 0 ↔ u =ᶠ[l] 0 := by |
rw [IsEquivalent, sub_zero]
exact isLittleO_zero_right_iff
|
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.Polynomial.Eval
#align_import data.mv_polynomial.polynomial from "leanprover-community/mathlib"@"0b8... | Mathlib/Algebra/MvPolynomial/Polynomial.lean | 19 | 28 | theorem polynomial_eval_eval₂ [CommSemiring R] [CommSemiring S]
{x : S} (f : R →+* Polynomial S) (g : σ → Polynomial S) (p : MvPolynomial σ R) :
Polynomial.eval x (eval₂ f g p) =
eval₂ ((Polynomial.evalRingHom x).comp f) (fun s => Polynomial.eval x (g s)) p := by |
apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp]
|
/-
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.RingTheory.FiniteType
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.finite... | Mathlib/RingTheory/FinitePresentation.lean | 532 | 535 | theorem of_surjective (f : A →ₐ[R] B) (hf : Surjective f) (hker : f.toRingHom.ker.FG) :
f.FinitePresentation := by |
-- Porting note: added `convert`
convert RingHom.FinitePresentation.of_surjective f hf hker
|
/-
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.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
#align_import linear_algebra.tensor_product... | Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 49 | 53 | theorem Matrix.toLin_kronecker (A : Matrix ι' ι R) (B : Matrix κ' κ R) :
toLin (bM.tensorProduct bN) (bM'.tensorProduct bN') (A ⊗ₖ B) =
TensorProduct.map (toLin bM bM' A) (toLin bN bN' B) := by |
rw [← LinearEquiv.eq_symm_apply, toLin_symm, TensorProduct.toMatrix_map, toMatrix_toLin,
toMatrix_toLin]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Devon Tuma
-/
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptoti... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 172 | 182 | theorem div_tendsto_atTop_of_degree_gt' (hdeg : Q.degree < P.degree)
(hpos : 0 < P.leadingCoeff / Q.leadingCoeff) :
Tendsto (fun x => eval x P / eval x Q) atTop atTop := by |
have hQ : Q ≠ 0 := fun h => by
simp only [h, div_zero, leadingCoeff_zero] at hpos
exact hpos.false
rw [← natDegree_lt_natDegree_iff hQ] at hdeg
refine (isEquivalent_atTop_div P Q).symm.tendsto_atTop ?_
apply Tendsto.const_mul_atTop hpos
apply tendsto_zpow_atTop_atTop
omega
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b6093548394... | Mathlib/SetTheory/Ordinal/Principal.lean | 186 | 201 | theorem add_omega_opow {a b : Ordinal} (h : a < (omega^b)) : a + (omega^b) = (omega^b) := by |
refine le_antisymm ?_ (le_add_left _ _)
induction' b using limitRecOn with b _ b l IH
· rw [opow_zero, ← succ_zero, lt_succ_iff, Ordinal.le_zero] at h
rw [h, zero_add]
· rw [opow_succ] at h
rcases (lt_mul_of_limit omega_isLimit).1 h with ⟨x, xo, ax⟩
refine le_trans (add_le_add_right (le_of_lt ax) _... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 682 | 689 | theorem Filter.HasBasis.uniformEquicontinuous_iff_left {p : κ → Prop}
{s : κ → Set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).HasBasis p s) :
UniformEquicontinuous F ↔
∀ U ∈ 𝓤 α, ∃ k, p k ∧ ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U := by |
rw [uniformEquicontinuous_iff_uniformContinuous, UniformContinuous,
hβ.tendsto_iff (UniformFun.hasBasis_uniformity ι α)]
simp only [Prod.forall]
rfl
|
/-
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.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Topology.UniformSpace.Compact
import Mathl... | Mathlib/Analysis/BoxIntegral/Basic.lean | 502 | 511 | theorem tendsto_integralSum_toFilter_prod_self_inf_iUnion_eq_uniformity (h : Integrable I l f vol) :
Tendsto (fun π : TaggedPrepartition I × TaggedPrepartition I =>
(integralSum f vol π.1, integralSum f vol π.2))
((l.toFilter I ×ˢ l.toFilter I) ⊓ 𝓟 {π | π.1.iUnion = π.2.iUnion}) (𝓤 F) := by |
refine (((l.hasBasis_toFilter I).prod_self.inf_principal _).tendsto_iff
uniformity_basis_dist_le).2 fun ε ε0 => ?_
replace ε0 := half_pos ε0
use h.convergenceR (ε / 2), h.convergenceR_cond (ε / 2); rintro ⟨π₁, π₂⟩ ⟨⟨h₁, h₂⟩, hU⟩
rw [← add_halves ε]
exact h.dist_integralSum_le_of_memBaseSet ε0 ε0 h₁.choos... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Grou... | Mathlib/RingTheory/Coprime/Basic.lean | 227 | 229 | theorem IsCoprime.of_mul_add_left_right (h : IsCoprime x (x * z + y)) : IsCoprime x y := by |
rw [add_comm] at h
exact h.of_add_mul_left_right
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.eq... | Mathlib/Order/Partition/Equipartition.lean | 180 | 182 | theorem indiscrete_isEquipartition {hs : s ≠ ∅} : (indiscrete hs).IsEquipartition := by |
rw [IsEquipartition, indiscrete_parts, coe_singleton]
exact Set.equitableOn_singleton s _
|
/-
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.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 517 | 519 | theorem ennrealToMeasure_apply {m : MeasurableSpace α} {v : VectorMeasure α ℝ≥0∞} {s : Set α}
(hs : MeasurableSet s) : ennrealToMeasure v s = v s := by |
rw [ennrealToMeasure, ofMeasurable_apply _ hs]
|
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import... | Mathlib/SetTheory/Surreal/Dyadic.lean | 52 | 52 | theorem powHalf_leftMoves (n) : (powHalf n).LeftMoves = PUnit := by | cases n <;> rfl
|
/-
Copyright (c) 2022 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Tactic.NoncommRing
#align_import... | Mathlib/Analysis/NormedSpace/MStructure.lean | 105 | 144 | theorem commute [FaithfulSMul M X] {P Q : M} (h₁ : IsLprojection X P) (h₂ : IsLprojection X Q) :
Commute P Q := by |
have PR_eq_RPR : ∀ R : M, IsLprojection X R → P * R = R * P * R := fun R h₃ => by
-- Porting note: Needed to fix function, which changes indent of following lines
refine @eq_of_smul_eq_smul _ X _ _ _ _ fun x => by
rw [← norm_sub_eq_zero_iff]
have e1 : ‖R • x‖ ≥ ‖R • x‖ + 2 • ‖(P * R) • x - (R * P... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 219 | 220 | theorem encard_diff_add_encard (s t : Set α) : (s \ t).encard + t.encard = (s ∪ t).encard := by |
rw [← encard_union_eq disjoint_sdiff_left, diff_union_self]
|
/-
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.Init.Align
import Mathlib.Topology.PartialHomeomorph
#align_import geometry.manifold.charted_space from "leanprover-community/mathlib"@"431589bc... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 714 | 723 | theorem ChartedSpace.locallyConnectedSpace [LocallyConnectedSpace H] : LocallyConnectedSpace M := by |
let e : M → PartialHomeomorph M H := chartAt H
refine locallyConnectedSpace_of_connected_bases (fun x s ↦ (e x).symm '' s)
(fun x s ↦ (IsOpen s ∧ e x x ∈ s ∧ IsConnected s) ∧ s ⊆ (e x).target) ?_ ?_
· intro x
simpa only [e, PartialHomeomorph.symm_map_nhds_eq, mem_chart_source] using
((LocallyConn... |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 38 | 40 | theorem mapAccumr_map (f₂ : α → β) :
(mapAccumr f₁ (map f₂ xs) s) = (mapAccumr (fun x s => f₁ (f₂ x) s) xs s) := by |
induction xs using Vector.revInductionOn generalizing s <;> simp_all
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 585 | 587 | theorem condexpL1_smul (c : 𝕜) (f : α → F') : condexpL1 hm μ (c • f) = c • condexpL1 hm μ f := by |
refine setToFun_smul _ ?_ c f
exact fun c _ x => condexpInd_smul' c x
|
/-
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.Basic
import Mathlib.Topology.Order.ProjIcc
#al... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 418 | 419 | theorem arccos_of_le_neg_one {x : ℝ} (hx : x ≤ -1) : arccos x = π := by |
rw [arccos, arcsin_of_le_neg_one hx, sub_neg_eq_add, add_halves']
|
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.LatticeIntervals
import Mathlib.Order.Interval.Set.OrdConnected
#align_import order.co... | Mathlib/Order/CompleteLatticeIntervals.lean | 106 | 108 | theorem subset_sInf_of_not_bddBelow [Inhabited s] {t : Set s} (ht : ¬BddBelow t) :
sInf t = default := by |
simp [sInf, ht]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 677 | 678 | theorem surjOn_cos : SurjOn cos (Icc 0 π) (Icc (-1) 1) := by |
simpa only [cos_zero, cos_pi] using intermediate_value_Icc' pi_pos.le continuous_cos.continuousOn
|
/-
Copyright (c) 2024 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
/-!
# Covolume of ℤ-lattices
Let `E` be a finite dimensional real vector space with an inner product.
Let `L` be a `ℤ`-l... | Mathlib/Algebra/Module/Zlattice/Covolume.lean | 58 | 62 | theorem covolume_ne_zero : covolume L μ ≠ 0 := by |
rw [covolume_eq_measure_fundamentalDomain L μ (isAddFundamentalDomain (Free.chooseBasis ℤ L) μ),
ENNReal.toReal_ne_zero]
refine ⟨Zspan.measure_fundamentalDomain_ne_zero _, ne_of_lt ?_⟩
exact Bornology.IsBounded.measure_lt_top (Zspan.fundamentalDomain_isBounded _)
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Generator.lean | 117 | 126 | theorem isDetecting_op_iff (𝒢 : Set C) : IsDetecting 𝒢.op ↔ IsCodetecting 𝒢 := by |
refine ⟨fun h𝒢 X Y f hf => ?_, fun h𝒢 X Y f hf => ?_⟩
· refine (isIso_op_iff _).1 (h𝒢 _ fun G hG h => ?_)
obtain ⟨t, ht, ht'⟩ := hf (unop G) (Set.mem_op.1 hG) h.unop
exact
⟨t.op, Quiver.Hom.unop_inj ht, fun y hy => Quiver.Hom.unop_inj (ht' _ (Quiver.Hom.op_inj hy))⟩
· refine (isIso_unop_iff _).1... |
/-
Copyright (c) 2019 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jannis Limperg, Mario Carneiro
-/
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
namespace Batteries
namespace BinomialHeap
namespac... | .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | 449 | 459 | theorem Heap.WF.deleteMin {s : Heap α}
(h : s.WF le n) (eq : s.deleteMin le = some (a, s')) : s'.WF le 0 := by |
cases s with cases eq | cons r a c s => ?_
have : (s.findMin le (cons r a c) ⟨id, a, c, s⟩).WF le :=
let ⟨_, h₂, h₃⟩ := h
h₃.findMin ⟨_, fun h => h.of_le (Nat.zero_le _), h₂, h₃⟩
fun h => ⟨Nat.zero_le _, h₂, h⟩
revert this
let { before, val, node, next } := s.findMin le (cons r a c) ⟨id, a, c, s⟩... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 120 | 120 | theorem log_neg_I : log (-I) = -(π / 2) * I := by | simp [log]
|
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 347 | 354 | theorem mul_div_mul_comm_of_dvd_dvd {a b c d : R} (hac : c ∣ a) (hbd : d ∣ b) :
a * b / (c * d) = a / c * (b / d) := by |
rcases eq_or_ne c 0 with (rfl | hc0); · simp
rcases eq_or_ne d 0 with (rfl | hd0); · simp
obtain ⟨k1, rfl⟩ := hac
obtain ⟨k2, rfl⟩ := hbd
rw [mul_div_cancel_left₀ _ hc0, mul_div_cancel_left₀ _ hd0, mul_mul_mul_comm,
mul_div_cancel_left₀ _ (mul_ne_zero hc0 hd0)]
|
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.rig... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 259 | 263 | theorem norm_div_tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle (x + y) y) = ‖y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).norm_div_tan_oangle_add_right_of_oangle_eq_pi_div_two h
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.M... | Mathlib/Analysis/Convex/Integral.lean | 286 | 309 | theorem StrictConvexOn.ae_eq_const_or_map_average_lt [IsFiniteMeasure μ] (hg : StrictConvexOn ℝ s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s) (hfi : Integrable f μ)
(hgi : Integrable (g ∘ f) μ) :
f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ g (⨍ x, f x ∂μ) < ⨍ x, g (f x) ∂μ := by |
have : ∀ {t}, μ t ≠ 0 → (⨍ x in t, f x ∂μ) ∈ s ∧ g (⨍ x in t, f x ∂μ) ≤ ⨍ x in t, g (f x) ∂μ :=
fun ht =>
hg.convexOn.set_average_mem_epigraph hgc hsc ht (measure_ne_top _ _) (ae_restrict_of_ae hfs)
hfi.integrableOn hgi.integrableOn
refine (ae_eq_const_or_exists_average_ne_compl hfi).imp_right ?_
r... |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymp... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 151 | 154 | theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) := by |
rcases em <| c = 0 with rfl | h
· exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
· exact (isEquivalent_const_iff_tendsto h).mp hu
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 2,034 | 2,051 | theorem contSupp_supports {S k} (H : contSupp k ⊆ S) : Supports (contSupp k) S := by |
induction k with
| halt => simp [contSupp_halt, Supports]
| cons₁ f k IH =>
have H₁ := H; rw [contSupp_cons₁] at H₁; have H₂ := Finset.union_subset_right H₁
refine trStmts₁_supports' (trNormal_supports H₂) H₁ fun h => ?_
refine supports_union.2 ⟨codeSupp'_supports H₂, ?_⟩
simp only [codeSupp, con... |
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 760 | 762 | theorem inv_smul_integral_comp_div (c) :
(c⁻¹ • ∫ x in a..b, f (x / c)) = ∫ x in a / c..b / c, f x := by |
by_cases hc : c = 0 <;> simp [hc, integral_comp_div]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 120 | 129 | theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 :=
suffices ∀ A B : Matrix n n α, A * B = 1 → B * A = 1 from ⟨this A B, this B A⟩
fun A B h => by
letI : Invertible B.det := detInvertibleOfLeftInverse _ _ h
letI : Invertible B := invertibleOfDetInvertible B
calc
B * A = B * A * (B * ⅟ B) := by | rw [mul_invOf_self, Matrix.mul_one]
_ = B * (A * B * ⅟ B) := by simp only [Matrix.mul_assoc]
_ = B * ⅟ B := by rw [h, Matrix.one_mul]
_ = 1 := mul_invOf_self B
|
/-
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.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-communi... | Mathlib/MeasureTheory/Group/Prod.lean | 325 | 334 | theorem measure_lintegral_div_measure [IsMulLeftInvariant ν] (sm : MeasurableSet s) (h2s : ν s ≠ 0)
(h3s : ν s ≠ ∞) (f : G → ℝ≥0∞) (hf : Measurable f) :
(μ s * ∫⁻ y, f y⁻¹ / ν ((fun x => x * y⁻¹) ⁻¹' s) ∂ν) = ∫⁻ x, f x ∂μ := by |
set g := fun y => f y⁻¹ / ν ((fun x => x * y⁻¹) ⁻¹' s)
have hg : Measurable g :=
(hf.comp measurable_inv).div ((measurable_measure_mul_right ν sm).comp measurable_inv)
simp_rw [measure_mul_lintegral_eq μ ν sm g hg, g, inv_inv]
refine lintegral_congr_ae ?_
refine (ae_measure_preimage_mul_right_lt_top_of_n... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 815 | 818 | theorem aemeasurable_iSup {ι} {μ : Measure δ} [Countable ι] {f : ι → δ → α}
(hf : ∀ i, AEMeasurable (f i) μ) : AEMeasurable (fun b => ⨆ i, f i b) μ := by |
refine ⟨fun b ↦ ⨆ i, (hf i).mk (f i) b, measurable_iSup (fun i ↦ (hf i).measurable_mk), ?_⟩
filter_upwards [ae_all_iff.2 (fun i ↦ (hf i).ae_eq_mk)] with b hb using by simp [hb]
|
/-
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.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 605 | 608 | theorem connectedComponentIn_nonempty_iff {x : α} {F : Set α} :
(connectedComponentIn F x).Nonempty ↔ x ∈ F := by |
rw [connectedComponentIn]
split_ifs <;> simp [connectedComponent_nonempty, *]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.LinearAlgebra.Dual
import Math... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 80 | 85 | theorem _root_.separatingDual_iff_injective : SeparatingDual R V ↔
Function.Injective (ContinuousLinearMap.coeLM (R := R) R (M := V) (N₃ := R)).flip := by |
simp_rw [separatingDual_def, Ne, injective_iff_map_eq_zero]
congrm ∀ v, ?_
rw [not_imp_comm, LinearMap.ext_iff]
push_neg; rfl
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 521 | 524 | theorem Convex.add_smul_sub_mem (h : Convex 𝕜 s) {x y : E} (hx : x ∈ s) (hy : y ∈ s) {t : 𝕜}
(ht : t ∈ Icc (0 : 𝕜) 1) : x + t • (y - x) ∈ s := by |
rw [add_comm]
exact h.lineMap_mem hx hy ht
|
/-
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, Mario Carneiro
-/
import Batteries.Tactic.Alias
import Batteries.Data.List.Init.Attach
import Batteries.Data.List.Pairwise
-- Adaptation note: ... | .lake/packages/batteries/Batteries/Data/List/Perm.lean | 528 | 532 | theorem cons_perm_iff_perm_erase {a : α} {l₁ l₂ : List α} :
a :: l₁ ~ l₂ ↔ a ∈ l₂ ∧ l₁ ~ l₂.erase a := by |
refine ⟨fun h => ?_, fun ⟨m, h⟩ => (h.cons a).trans (perm_cons_erase m).symm⟩
have : a ∈ l₂ := h.subset (mem_cons_self a l₁)
exact ⟨this, (h.trans <| perm_cons_erase this).cons_inv⟩
|
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