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) 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.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e31... | Mathlib/Data/QPF/Multivariate/Basic.lean | 282 | 283 | theorem liftpPreservation_iff_uniform : q.LiftPPreservation ↔ q.IsUniform := by |
rw [← suppPreservation_iff_liftpPreservation, suppPreservation_iff_isUniform]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 499 | 501 | theorem sup_coe_toSubmodule :
(↑(N ⊔ N') : Submodule R M) = (N : Submodule R M) ⊔ (N' : Submodule R M) := by |
rfl
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 229 | 253 | theorem condexpL2_const_inner (hm : m ≤ m0) (f : Lp E 2 μ) (c : E) :
condexpL2 𝕜 𝕜 hm (((Lp.memℒp f).const_inner c).toLp fun a => ⟪c, f a⟫) =ᵐ[μ]
fun a => ⟪c, (condexpL2 E 𝕜 hm f : α → E) a⟫ := by |
rw [lpMeas_coe]
have h_mem_Lp : Memℒp (fun a => ⟪c, (condexpL2 E 𝕜 hm f : α → E) a⟫) 2 μ := by
refine Memℒp.const_inner _ ?_; rw [lpMeas_coe]; exact Lp.memℒp _
have h_eq : h_mem_Lp.toLp _ =ᵐ[μ] fun a => ⟪c, (condexpL2 E 𝕜 hm f : α → E) a⟫ :=
h_mem_Lp.coeFn_toLp
refine EventuallyEq.trans ?_ h_eq
ref... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601... | Mathlib/Data/List/NatAntidiagonal.lean | 52 | 53 | theorem length_antidiagonal (n : ℕ) : (antidiagonal n).length = n + 1 := by |
rw [antidiagonal, length_map, length_range]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 1,118 | 1,119 | theorem single_add_erase (a : α) (f : α →₀ M) : single a (f a) + f.erase a = f := by |
rw [← update_eq_single_add_erase, update_self]
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Data.Set.Function
#align_import analysis.sum_integral_comparisons from "leanprover-community/... | Mathlib/Analysis/SumIntegralComparisons.lean | 162 | 165 | theorem MonotoneOn.integral_le_sum (hf : MonotoneOn f (Icc x₀ (x₀ + a))) :
(∫ x in x₀..x₀ + a, f x) ≤ ∑ i ∈ Finset.range a, f (x₀ + (i + 1 : ℕ)) := by |
rw [← neg_le_neg_iff, ← Finset.sum_neg_distrib, ← intervalIntegral.integral_neg]
exact hf.neg.sum_le_integral
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 366 | 373 | theorem degree_interpolate_lt (hvs : Set.InjOn v s) : (interpolate s v r).degree < s.card := by |
rw [Nat.cast_withBot]
rcases eq_empty_or_nonempty s with (rfl | h)
· rw [interpolate_empty, degree_zero, card_empty]
exact WithBot.bot_lt_coe _
· refine lt_of_le_of_lt (degree_interpolate_le _ hvs) ?_
rw [Nat.cast_withBot, WithBot.coe_lt_coe]
exact Nat.sub_lt (Nonempty.card_pos h) zero_lt_one
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 171 | 177 | theorem formPerm_apply_get_zero (l : List α) (h : Nodup l) (hl : 1 < l.length) :
formPerm l (l.get (Fin.mk 0 (by omega))) = l.get (Fin.mk 1 hl) := by |
rcases l with (_ | ⟨x, _ | ⟨y, tl⟩⟩)
· simp at hl
· rw [get, get_singleton]; rfl;
· rw [get, formPerm_apply_head, get, get]
exact h
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267... | Mathlib/Data/Finset/Prod.lean | 244 | 246 | theorem union_product [DecidableEq α] [DecidableEq β] : (s ∪ s') ×ˢ t = s ×ˢ t ∪ s' ×ˢ t := by |
ext ⟨x, y⟩
simp only [or_and_right, mem_union, mem_product]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic... | Mathlib/GroupTheory/Perm/Basic.lean | 428 | 432 | theorem subtypePerm_zpow (f : Perm α) (n : ℤ) (hf) :
(f.subtypePerm hf ^ n : Perm { x // p x }) = (f ^ n).subtypePerm (zpow_aux hf) := by |
induction' n with n ih
· exact subtypePerm_pow _ _ _
· simp only [zpow_negSucc, subtypePerm_pow, subtypePerm_inv]
|
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 538 | 539 | theorem insert_singleton_eq {a : α} {b b' : β a} : insert a b (singleton a b') = singleton a b := by |
simp only [singleton, Finmap.insert_toFinmap, AList.insert_singleton_eq]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 782 | 790 | theorem nonneg_smul {a : ℤ√d} {n : ℕ} (ha : Nonneg a) : Nonneg ((n : ℤ√d) * a) := by |
rw [← Int.cast_natCast n]
exact
match a, nonneg_cases ha, ha with
| _, ⟨x, y, Or.inl rfl⟩, _ => by rw [smul_val]; trivial
| _, ⟨x, y, Or.inr <| Or.inl rfl⟩, ha => by
rw [smul_val]; simpa using nonnegg_pos_neg.2 (sqLe_smul n <| nonnegg_pos_neg.1 ha)
| _, ⟨x, y, Or.inr <| Or.inr rfl⟩, ha => by
... |
/-
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 | 661 | 663 | theorem integral_const (c : E) : ∫ _ in a..b, c = (b - a) • c := by |
simp only [integral_const', Real.volume_Ioc, ENNReal.toReal_ofReal', ← neg_sub b,
max_zero_sub_eq_self]
|
/-
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 | 1,276 | 1,278 | theorem degree_linear_lt_degree_C_mul_X_sq (ha : a ≠ 0) :
degree (C b * X + C c) < degree (C a * X ^ 2) := by |
simpa only [degree_C_mul_X_pow 2 ha] using degree_linear_lt
|
/-
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.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 50 | 54 | theorem Nat.Coprime.cast {R : Type*} [CommRing R] {a b : ℕ} (h : Nat.Coprime a b) :
IsCoprime (a : R) (b : R) := by |
rw [← isCoprime_iff_coprime] at h
rw [← Int.cast_natCast a, ← Int.cast_natCast b]
exact IsCoprime.intCast h
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 374 | 375 | theorem pow_apply_mem_support {n : ℕ} {x : α} : (f ^ n) x ∈ f.support ↔ x ∈ f.support := by |
simp only [mem_support, ne_eq, apply_pow_apply_eq_iff]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92... | Mathlib/LinearAlgebra/LinearPMap.lean | 882 | 885 | theorem mem_domain_of_mem_graph {f : E →ₗ.[R] F} {x : E} {y : F} (h : (x, y) ∈ f.graph) :
x ∈ f.domain := by |
rw [mem_domain_iff]
exact ⟨y, h⟩
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 809 | 812 | theorem exists_not_mem_null_lintegral_le (hint : ∫⁻ a, f a ∂μ ≠ ∞) (hN : μ N = 0) :
∃ x, x ∉ N ∧ ∫⁻ a, f a ∂μ ≤ f x := by |
simpa only [laverage_eq_lintegral] using
exists_not_mem_null_laverage_le (IsProbabilityMeasure.ne_zero μ) hint hN
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 497 | 507 | theorem circumcenter_eq_of_range_eq {n : ℕ} {s₁ s₂ : Simplex ℝ P n}
(h : Set.range s₁.points = Set.range s₂.points) : s₁.circumcenter = s₂.circumcenter := by |
have hs : s₁.circumcenter ∈ affineSpan ℝ (Set.range s₂.points) :=
h ▸ s₁.circumcenter_mem_affineSpan
have hr : ∀ i, dist (s₂.points i) s₁.circumcenter = s₁.circumradius := by
intro i
have hi : s₂.points i ∈ Set.range s₂.points := Set.mem_range_self _
rw [← h, Set.mem_range] at hi
rcases hi with... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic fr... | Mathlib/Topology/MetricSpace/Basic.lean | 191 | 193 | theorem MetricSpace.replaceUniformity_eq {γ} [U : UniformSpace γ] (m : MetricSpace γ)
(H : 𝓤[U] = 𝓤[PseudoEMetricSpace.toUniformSpace]) : m.replaceUniformity H = m := by |
ext; rfl
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison
-/
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import ... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 401 | 403 | theorem const_mul_cancel (f g₁ g₂ : R) (U hu₁ hu₂) :
const R f g₁ U hu₁ * const R g₁ g₂ U hu₂ = const R f g₂ U hu₂ := by |
rw [const_mul, const_ext]; rw [mul_assoc]
|
/-
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.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 426 | 433 | theorem iUnion_ofWithBot (boxes : Finset (WithBot (Box ι)))
(le_of_mem : ∀ J ∈ boxes, (J : WithBot (Box ι)) ≤ I)
(pairwise_disjoint : Set.Pairwise (boxes : Set (WithBot (Box ι))) Disjoint) :
(ofWithBot boxes le_of_mem pairwise_disjoint).iUnion = ⋃ J ∈ boxes, ↑J := by |
suffices ⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J = ⋃ J ∈ boxes, (J : Set (ι → ℝ)) by
simpa [ofWithBot, Prepartition.iUnion]
simp only [← Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box ι), @iUnion_comm _ _ (@Eq _ _ _),
iUnion_iUnion_eq_right]
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Car... | Mathlib/Combinatorics/Configuration.lean | 467 | 493 | theorem card_points [Fintype P] [Finite L] : Fintype.card P = order P L ^ 2 + order P L + 1 := by |
cases nonempty_fintype L
obtain ⟨p, -⟩ := @exists_config P L _ _
let ϕ : { q // q ≠ p } ≃ Σl : { l : L // p ∈ l }, { q // q ∈ l.1 ∧ q ≠ p } :=
{ toFun := fun q => ⟨⟨mkLine q.2, (mkLine_ax q.2).2⟩, q, (mkLine_ax q.2).1, q.2⟩
invFun := fun lq => ⟨lq.2, lq.2.2.2⟩
left_inv := fun q => Subtype.ext rfl... |
/-
Copyright (c) 2021 Roberto Alvarez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Roberto Alvarez
-/
import Mathlib.AlgebraicTopology.FundamentalGroupoid.FundamentalGroup
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Logic.Equiv.TransferInstance
import Ma... | Mathlib/Topology/Homotopy/HomotopyGroup.lean | 553 | 557 | theorem inv_spec [Nonempty N] {i} {p : Ω^ N X x} :
((⟦p⟧)⁻¹ : HomotopyGroup N X x) = ⟦symmAt i p⟧ := by |
rw [symmAt_indep (Classical.arbitrary N) p, ← fromLoop_symm_toLoop]
apply Quotient.sound
rfl
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Kernels.lean | 138 | 140 | theorem PreservesKernel.iso_hom : (PreservesKernel.iso G f).hom = kernelComparison f G := by |
rw [← cancel_mono (kernel.ι _)]
simp [PreservesKernel.iso]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Ring/Center.lean | 81 | 86 | theorem neg_mem_center [NonUnitalNonAssocRing M] {a : M} (ha : a ∈ Set.center M) :
-a ∈ Set.center M where
comm _ := by | rw [← neg_mul_comm, ← ha.comm, neg_mul_comm]
left_assoc _ _ := by rw [neg_mul, ha.left_assoc, neg_mul, neg_mul]
mid_assoc _ _ := by rw [← neg_mul_comm, ha.mid_assoc, neg_mul_comm, neg_mul]
right_assoc _ _ := by rw [mul_neg, ha.right_assoc, mul_neg, mul_neg]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 372 | 390 | theorem rcons_inj {i} : Function.Injective (rcons : Pair M i → Word M) := by |
rintro ⟨m, w, h⟩ ⟨m', w', h'⟩ he
by_cases hm : m = 1 <;> by_cases hm' : m' = 1
· simp only [rcons, dif_pos hm, dif_pos hm'] at he
aesop
· exfalso
simp only [rcons, dif_pos hm, dif_neg hm'] at he
rw [he] at h
exact h rfl
· exfalso
simp only [rcons, dif_pos hm', dif_neg hm] at he
rw [← ... |
/-
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 | 522 | 529 | theorem isSeparator_iff_epi (G : C) [∀ A : C, HasCoproduct fun _ : G ⟶ A => G] :
IsSeparator G ↔ ∀ A : C, Epi (Sigma.desc fun f : G ⟶ A => f) := by |
rw [isSeparator_def]
refine ⟨fun h A => ⟨fun u v huv => h _ _ fun i => ?_⟩, fun h X Y f g hh => ?_⟩
· simpa using Sigma.ι _ i ≫= huv
· haveI := h X
refine (cancel_epi (Sigma.desc fun f : G ⟶ X => f)).1 (colimit.hom_ext fun j => ?_)
simpa using hh j.as
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 88 | 92 | theorem normalize_num_den' (num den nz) : ∃ d : Nat, d ≠ 0 ∧
num = (normalize num den nz).num * d ∧ den = (normalize num den nz).den * d := by |
refine ⟨num.natAbs.gcd den, Nat.gcd_ne_zero_right nz, ?_⟩
simp [normalize_eq, Int.ediv_mul_cancel (Int.ofNat_dvd_left.2 <| Nat.gcd_dvd_left ..),
Nat.div_mul_cancel (Nat.gcd_dvd_right ..)]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorS... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 119 | 123 | theorem mem_restrictDegree_iff_sup [DecidableEq σ] (p : MvPolynomial σ R) (n : ℕ) :
p ∈ restrictDegree σ R n ↔ ∀ i, p.degrees.count i ≤ n := by |
simp only [mem_restrictDegree, degrees_def, Multiset.count_finset_sup, Finsupp.count_toMultiset,
Finset.sup_le_iff]
exact ⟨fun h n s hs => h s hs n, fun h s hs n => h n s hs⟩
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yourong Zang
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mat... | Mathlib/Analysis/Complex/RealDeriv.lean | 68 | 81 | theorem HasDerivAt.real_of_complex (h : HasDerivAt e e' z) :
HasDerivAt (fun x : ℝ => (e x).re) e'.re z := by |
have A : HasFDerivAt ((↑) : ℝ → ℂ) ofRealCLM z := ofRealCLM.hasFDerivAt
have B :
HasFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)
(ofRealCLM z) :=
h.hasFDerivAt.restrictScalars ℝ
have C : HasFDerivAt re reCLM (e (ofRealCLM z)) := reCLM.hasFDerivAt
-- Porting note... |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.AList
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Part
#align_import data.finmap from "leanprover-community/mathlib"@"c... | Mathlib/Data/Finmap.lean | 370 | 372 | theorem replace_toFinmap (a : α) (b : β a) (s : AList β) :
replace a b ⟦s⟧ = (⟦s.replace a b⟧ : Finmap β) := by |
simp [replace]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 514 | 516 | theorem iSup_coe_toSubmodule {ι} (p : ι → LieSubmodule R L M) :
(↑(⨆ i, p i) : Submodule R M) = ⨆ i, (p i : Submodule R M) := by |
rw [iSup, sSup_coe_toSubmodule]; ext; simp [Submodule.mem_sSup, Submodule.mem_iSup]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 172 | 173 | theorem rotate_length_mul (l : List α) (n : ℕ) : l.rotate (l.length * n) = l := by |
rw [rotate_eq_rotate', rotate'_length_mul]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathli... | Mathlib/CategoryTheory/Extensive.lean | 407 | 419 | theorem finitaryExtensive_of_preserves_and_reflects (F : C ⥤ D) [FinitaryExtensive D]
[HasFiniteCoproducts C] [HasPullbacksOfInclusions C]
[PreservesPullbacksOfInclusions F]
[ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]
[ReflectsColimitsOfShape (Discrete Wal... |
constructor
intros X Y c hc
refine IsVanKampenColimit.of_iso ?_ (hc.uniqueUpToIso (coprodIsCoprod X Y)).symm
have (i : Discrete WalkingPair) (Z : C) (f : Z ⟶ X ⨿ Y) :
PreservesLimit (cospan f ((BinaryCofan.mk coprod.inl coprod.inr).ι.app i)) F := by
rcases i with ⟨_|_⟩ <;> dsimp <;> infer_instance
re... |
/-
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 | 232 | 315 | theorem of_restrict_scalars_finitePresentation [Algebra A B] [IsScalarTower R A B]
[FinitePresentation.{w₁, w₃} R B] [FiniteType R A] :
FinitePresentation.{w₂, w₃} A B := by |
classical
obtain ⟨n, f, hf, s, hs⟩ := FinitePresentation.out (R := R) (A := B)
letI RX := MvPolynomial (Fin n) R
letI AX := MvPolynomial (Fin n) A
refine ⟨n, MvPolynomial.aeval (f ∘ X), ?_, ?_⟩
· rw [← Algebra.range_top_iff_surjective, ← Algebra.adjoin_range_eq_range_aeval,
Set.range_comp f MvPolynom... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
/-!
# ... | Mathlib/Topology/Baire/Lemmas.lean | 74 | 81 | theorem mem_residual {s : Set X} : s ∈ residual X ↔ ∃ t ⊆ s, IsGδ t ∧ Dense t := by |
constructor
· rw [mem_residual_iff]
rintro ⟨S, hSo, hSd, Sct, Ss⟩
refine ⟨_, Ss, ⟨_, fun t ht => hSo _ ht, Sct, rfl⟩, ?_⟩
exact dense_sInter_of_isOpen hSo Sct hSd
rintro ⟨t, ts, ho, hd⟩
exact mem_of_superset (residual_of_dense_Gδ ho hd) ts
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 153 | 155 | theorem uniformEmbedding_iff' {f : α → β} :
UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧ comap (Prod.map f f) (𝓤 β) ≤ 𝓤 α := by |
rw [uniformEmbedding_iff, and_comm, uniformInducing_iff']
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
#align_import data.pnat.prime from "leanprover-community/mathlib"@"09597669f0242... | Mathlib/Data/PNat/Prime.lean | 186 | 189 | theorem coprime_coe {m n : ℕ+} : Nat.Coprime ↑m ↑n ↔ m.Coprime n := by |
unfold Nat.Coprime Coprime
rw [← coe_inj]
simp
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebr... | Mathlib/LinearAlgebra/Finsupp.lean | 234 | 234 | theorem lapply_comp_lsingle_same (a : α) : lapply a ∘ₗ lsingle a = (.id : M →ₗ[R] M) := by | ext; simp
|
/-
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.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCo... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 922 | 925 | theorem map_add_left (f₁ f₂ : M →ₗ[R] P) (g : N →ₗ[R] Q) :
map (f₁ + f₂) g = map f₁ g + map f₂ g := by |
ext
simp only [add_tmul, compr₂_apply, mk_apply, map_tmul, add_apply]
|
/-
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.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 891 | 891 | theorem sigma_zero_apply (n : ℕ) : σ 0 n = (divisors n).card := by | simp [sigma_apply]
|
/-
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
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 513 | 514 | theorem toReal_inv (a : ℝ≥0∞) : a⁻¹.toReal = a.toReal⁻¹ := by |
simp only [ENNReal.toReal, toNNReal_inv, NNReal.coe_inv]
|
/-
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.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 47 | 48 | theorem verschiebungFun_coeff_zero (x : 𝕎 R) : (verschiebungFun x).coeff 0 = 0 := by |
rw [verschiebungFun_coeff, if_pos rfl]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,294 | 1,295 | theorem image_mul_right' :
image (· * b⁻¹) t = preimage t (· * b) (mul_left_injective _).injOn := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
#align_import data.polynomial.splits from "leanpro... | Mathlib/Algebra/Polynomial/Splits.lean | 181 | 187 | theorem roots_ne_zero_of_splits' {f : K[X]} (hs : Splits i f) (hf0 : natDegree (f.map i) ≠ 0) :
(f.map i).roots ≠ 0 :=
let ⟨x, hx⟩ := exists_root_of_splits' i hs fun h => hf0 <| natDegree_eq_of_degree_eq_some h
fun h => by
rw [← eval_map] at hx
have : f.map i ≠ 0 := by | intro; simp_all
cases h.subst ((mem_roots this).2 hx)
|
/-
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, Kyle Miller
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align... | Mathlib/Data/Set/Finite.lean | 1,619 | 1,625 | theorem iUnion_pi_of_monotone {ι ι' : Type*} [LinearOrder ι'] [Nonempty ι'] {α : ι → Type*}
{I : Set ι} {s : ∀ i, ι' → Set (α i)} (hI : I.Finite) (hs : ∀ i ∈ I, Monotone (s i)) :
⋃ j : ι', I.pi (fun i => s i j) = I.pi fun i => ⋃ j, s i j := by |
simp only [pi_def, biInter_eq_iInter, preimage_iUnion]
haveI := hI.fintype.finite
refine iUnion_iInter_of_monotone (ι' := ι') (fun (i : I) j₁ j₂ h => ?_)
exact preimage_mono <| hs i i.2 h
|
/-
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, Sander Dahmen,
Scott Morrison, Chris Hughes, Anne Baanen, Junyan Xu
-/
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.Dimen... | Mathlib/LinearAlgebra/Dimension/DivisionRing.lean | 59 | 63 | theorem rank_quotient_add_rank_of_divisionRing (p : Submodule K V) :
Module.rank K (V ⧸ p) + Module.rank K p = Module.rank K V := by |
classical
let ⟨f⟩ := quotient_prod_linearEquiv p
exact rank_prod'.symm.trans f.rank_eq
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatoric... | Mathlib/Combinatorics/HalesJewett.lean | 211 | 212 | theorem diagonal_apply {α ι} [Nonempty ι] (x : α) : Line.diagonal α ι x = fun _ => x := by |
simp_rw [Line.diagonal, Option.getD_none]
|
/-
Copyright (c) 2021 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.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 500 | 503 | theorem gauge_eq_one_iff_mem_frontier (hc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) :
gauge s x = 1 ↔ x ∈ frontier s := by |
rw [eq_iff_le_not_lt, gauge_le_one_iff_mem_closure hc hs₀, gauge_lt_one_iff_mem_interior hc hs₀]
rfl
|
/-
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.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 694 | 698 | theorem LinearMap.toMatrix_comp [Finite l] [DecidableEq m] (f : M₂ →ₗ[R] M₃) (g : M₁ →ₗ[R] M₂) :
LinearMap.toMatrix v₁ v₃ (f.comp g) =
LinearMap.toMatrix v₂ v₃ f * LinearMap.toMatrix v₁ v₂ g := by |
simp_rw [LinearMap.toMatrix, LinearEquiv.trans_apply, LinearEquiv.arrowCongr_comp _ v₂.equivFun,
LinearMap.toMatrix'_comp]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 629 | 633 | theorem coe_ofInjective_symm {α β} {f : α → β} (hf : Injective f) :
((ofInjective f hf).symm : range f → α) = rangeSplitting f := by |
ext ⟨y, x, rfl⟩
apply hf
simp [apply_rangeSplitting f]
|
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ri... | Mathlib/RingTheory/Nilpotent/Defs.lean | 197 | 205 | theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S]
(f : F) (hf : Function.Injective f) [IsReduced S] :
IsReduced R := by |
constructor
intro x hx
apply hf
rw [map_zero]
exact (hx.map f).eq_zero
|
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 289 | 291 | theorem Filter.EventuallyEq.gradient_eq (hL : f₁ =ᶠ[𝓝 x] f) : ∇ f₁ x = ∇ f x := by |
unfold gradient
rwa [Filter.EventuallyEq.fderiv_eq]
|
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | Mathlib/Data/Nat/MaxPowDiv.lean | 46 | 54 | theorem go_succ {k p n : ℕ} : go (k+1) p n = go k p n + 1 := by |
induction k, p, n using go.induct
case case1 h ih =>
unfold go
simp only [if_pos h]
exact ih
case case2 h =>
unfold go
simp only [if_neg h]
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 274 | 274 | theorem aleph_zero : aleph 0 = ℵ₀ := by | rw [aleph, add_zero, aleph'_omega]
|
/-
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 | 1,287 | 1,288 | theorem ceil_sub_one (a : α) : ⌈a - 1⌉ = ⌈a⌉ - 1 := by |
rw [eq_sub_iff_add_eq, ← ceil_add_one, sub_add_cancel]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 605 | 605 | theorem orthogonalProjection_bot : orthogonalProjection (⊥ : Submodule 𝕜 E) = 0 := by | ext
|
/-
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.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 47 | 49 | theorem Ioc_filter_dvd_card : ((Ioc a b).filter (r ∣ ·)).card =
max (⌊b / (r : ℚ)⌋ - ⌊a / (r : ℚ)⌋) 0 := by |
rw [Ioc_filter_dvd_eq _ _ hr, card_map, card_Ioc, toNat_eq_max]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 881 | 884 | theorem HasBasis.tendsto_left_iff (hla : la.HasBasis pa sa) :
Tendsto f la lb ↔ ∀ t ∈ lb, ∃ i, pa i ∧ MapsTo f (sa i) t := by |
simp only [Tendsto, (hla.map f).le_iff, image_subset_iff]
rfl
|
/-
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.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import pr... | Mathlib/Probability/Martingale/Convergence.lean | 214 | 236 | theorem Submartingale.ae_tendsto_limitProcess [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, snorm (f n) 1 μ ≤ R) :
∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (ℱ.limitProcess f μ ω)) := by |
classical
suffices
∃ g, StronglyMeasurable[⨆ n, ℱ n] g ∧ ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop (𝓝 (g ω)) by
rw [limitProcess, dif_pos this]
exact (Classical.choose_spec this).2
set g' : Ω → ℝ := fun ω => if h : ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) then h.choose else 0
have hle : ⨆ n, ℱ n... |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_spac... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 83 | 105 | theorem riesz_lemma_of_norm_lt {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R) {F : Subspace 𝕜 E}
(hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) :
∃ x₀ : E, ‖x₀‖ ≤ R ∧ ∀ y ∈ F, 1 ≤ ‖x₀ - y‖ := by |
have Rpos : 0 < R := (norm_nonneg _).trans_lt hR
have : ‖c‖ / R < 1 := by
rw [div_lt_iff Rpos]
simpa using hR
rcases riesz_lemma hFc hF this with ⟨x, xF, hx⟩
have x0 : x ≠ 0 := fun H => by simp [H] at xF
obtain ⟨d, d0, dxlt, ledx, -⟩ :
∃ d : 𝕜, d ≠ 0 ∧ ‖d • x‖ < R ∧ R / ‖c‖ ≤ ‖d • x‖ ∧ ‖d‖⁻¹ ≤ R... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 214 | 216 | theorem degrees_map_of_injective [CommSemiring S] (p : MvPolynomial σ R) {f : R →+* S}
(hf : Injective f) : (map f p).degrees = p.degrees := by |
simp only [degrees, MvPolynomial.support_map_of_injective _ hf]
|
/-
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.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 810 | 837 | theorem contMDiffWithinAt_iff_contMDiffOn_nhds {n : ℕ} :
ContMDiffWithinAt I I' n f s x ↔ ∃ u ∈ 𝓝[insert x s] x, ContMDiffOn I I' n f u := by |
-- WLOG, `x ∈ s`, otherwise we add `x` to `s`
wlog hxs : x ∈ s generalizing s
· rw [← contMDiffWithinAt_insert_self, this (mem_insert _ _), insert_idem]
rw [insert_eq_of_mem hxs]
-- The `←` implication is trivial
refine ⟨fun h ↦ ?_, fun ⟨u, hmem, hu⟩ ↦ (hu _ (mem_of_mem_nhdsWithin hxs hmem)).mono_of_mem hm... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.le... | Mathlib/Data/Int/Lemmas.lean | 90 | 94 | theorem natAbs_coe_sub_coe_lt_of_lt {a b n : ℕ} (a_lt_n : a < n) (b_lt_n : b < n) :
natAbs (a - b : ℤ) < n := by |
rw [← Nat.cast_lt (α := ℤ), natCast_natAbs]
exact abs_sub_lt_of_nonneg_of_lt (ofNat_nonneg a) (ofNat_lt.mpr a_lt_n)
(ofNat_nonneg b) (ofNat_lt.mpr b_lt_n)
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 710 | 714 | theorem mem_keys_kunion {a} {l₁ l₂ : List (Sigma β)} :
a ∈ (kunion l₁ l₂).keys ↔ a ∈ l₁.keys ∨ a ∈ l₂.keys := by |
induction l₁ generalizing l₂ with
| nil => simp
| cons s l₁ ih => by_cases h : a = s.1 <;> [simp [h]; simp [h, ih]]
|
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#alig... | Mathlib/Algebra/Polynomial/Roots.lean | 69 | 73 | theorem card_roots (hp0 : p ≠ 0) : (Multiset.card (roots p) : WithBot ℕ) ≤ degree p := by |
classical
unfold roots
rw [dif_neg hp0]
exact (Classical.choose_spec (exists_multiset_roots hp0)).1
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 221 | 223 | theorem image_coe_Ioc : (some : α → WithBot α) '' Ioc a b = Ioc (a : WithBot α) b := by |
rw [← preimage_coe_Ioc, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Subset.trans Ioc_subset_Ioi_self <| Ioi_subset_Ioi bot_le)]
|
/-
Copyright (c) 2022 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.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.theta from "leanprover-community... | Mathlib/Analysis/Asymptotics/Theta.lean | 151 | 151 | theorem isTheta_norm_left : (fun x ↦ ‖f' x‖) =Θ[l] g ↔ f' =Θ[l] g := by | simp [IsTheta]
|
/-
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.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 687 | 693 | theorem mem_coclosedCompact_iff :
s ∈ coclosedCompact X ↔ IsCompact (closure sᶜ) := by |
refine hasBasis_coclosedCompact.mem_iff.trans ⟨?_, fun h ↦ ?_⟩
· rintro ⟨t, ⟨htcl, htco⟩, hst⟩
exact htco.of_isClosed_subset isClosed_closure <|
closure_minimal (compl_subset_comm.2 hst) htcl
· exact ⟨closure sᶜ, ⟨isClosed_closure, h⟩, compl_subset_comm.2 subset_closure⟩
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedb... | Mathlib/LinearAlgebra/Projection.lean | 211 | 215 | theorem linear_proj_add_linearProjOfIsCompl_eq_self (hpq : IsCompl p q) (x : E) :
(p.linearProjOfIsCompl q hpq x + q.linearProjOfIsCompl p hpq.symm x : E) = x := by |
dsimp only [linearProjOfIsCompl]
rw [← prodComm_trans_prodEquivOfIsCompl _ _ hpq]
exact (prodEquivOfIsCompl _ _ hpq).apply_symm_apply x
|
/-
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.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.L... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 150 | 158 | theorem abs_areaForm_of_orthogonal {x y : E} (h : ⟪x, y⟫ = 0) : |ω x y| = ‖x‖ * ‖y‖ := by |
rw [o.areaForm_to_volumeForm, o.abs_volumeForm_apply_of_pairwise_orthogonal]
· simp [Fin.prod_univ_succ]
intro i j hij
fin_cases i <;> fin_cases j
· simp_all
· simpa using h
· simpa [real_inner_comm] using h
· simp_all
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 109 | 110 | theorem derivative_X_pow (n : ℕ) : derivative (X ^ n : R[X]) = C (n : R) * X ^ (n - 1) := by |
convert derivative_C_mul_X_pow (1 : R) n <;> simp
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
#align_import topology.is_locally_homeomorph from "leanprover-community/mathlib"@"e9... | Mathlib/Topology/IsLocalHomeomorph.lean | 155 | 158 | theorem isLocalHomeomorph_iff_openEmbedding_restrict {f : X → Y} :
IsLocalHomeomorph f ↔ ∀ x : X, ∃ U ∈ 𝓝 x, OpenEmbedding (U.restrict f) := by |
simp_rw [isLocalHomeomorph_iff_isLocalHomeomorphOn_univ,
isLocalHomeomorphOn_iff_openEmbedding_restrict, imp_iff_right (Set.mem_univ _)]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingTheory.A... | Mathlib/RingTheory/AdjoinRoot.lean | 854 | 858 | theorem quotEquivQuotMap_apply_mk (f g : R[X]) (I : Ideal R) :
AdjoinRoot.quotEquivQuotMap f I (Ideal.Quotient.mk (Ideal.map (of f) I) (AdjoinRoot.mk f g)) =
Ideal.Quotient.mk (Ideal.span ({Polynomial.map (Ideal.Quotient.mk I) f} : Set (R ⧸ I)[X]))
(g.map (Ideal.Quotient.mk I)) := by |
rw [AdjoinRoot.quotEquivQuotMap_apply, AdjoinRoot.quotAdjoinRootEquivQuotPolynomialQuot_mk_of]
|
/-
Copyright (c) 2022 Matej Penciak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matej Penciak, Moritz Doll, Fabien Clery
-/
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.symplectic_group from "leanprover-community/mathlib"@"70... | Mathlib/LinearAlgebra/SymplecticGroup.lean | 52 | 55 | theorem J_squared : J l R * J l R = -1 := by |
rw [J, fromBlocks_multiply]
simp only [Matrix.zero_mul, Matrix.neg_mul, zero_add, neg_zero, Matrix.one_mul, add_zero]
rw [← neg_zero, ← Matrix.fromBlocks_neg, ← fromBlocks_one]
|
/-
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.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 236 | 253 | theorem comp_rightAdjointMate {X Y Z : C} [HasRightDual X] [HasRightDual Y] [HasRightDual Z]
{f : X ⟶ Y} {g : Y ⟶ Z} : (f ≫ g)ᘁ = gᘁ ≫ fᘁ := by |
rw [rightAdjointMate_comp]
simp only [rightAdjointMate, comp_whiskerRight]
simp only [← Category.assoc]; congr 3; simp only [Category.assoc]
simp only [← MonoidalCategory.whiskerLeft_comp]; congr 2
symm
calc
_ = 𝟙 _ ⊗≫ (η_ Y Yᘁ ▷ 𝟙_ C ≫ (Y ⊗ Yᘁ) ◁ η_ X Xᘁ) ⊗≫ Y ◁ Yᘁ ◁ f ▷ Xᘁ ⊗≫
Y ◁ ε_ Y Yᘁ ▷ ... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 221 | 223 | theorem iteratedDeriv_eq_equiv_comp : iteratedDeriv n f =
(ContinuousMultilinearMap.piFieldEquiv 𝕜 (Fin n) F).symm ∘ iteratedFDeriv 𝕜 n f := by |
ext x; rfl
|
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Basic
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a10... | Mathlib/Algebra/Group/Semiconj/Basic.lean | 26 | 27 | theorem inv_inv_symm_iff : SemiconjBy a⁻¹ x⁻¹ y⁻¹ ↔ SemiconjBy a y x := by |
simp_rw [SemiconjBy, ← mul_inv_rev, inv_inj, eq_comm]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 109 | 129 | theorem cardQuot_mul_of_coprime [Module.Free ℤ S] [Module.Finite ℤ S]
{I J : Ideal S} (coprime : IsCoprime I J) : cardQuot (I * J) = cardQuot I * cardQuot J := by |
let b := Module.Free.chooseBasis ℤ S
cases isEmpty_or_nonempty (Module.Free.ChooseBasisIndex ℤ S)
· haveI : Subsingleton S := Function.Surjective.subsingleton b.repr.toEquiv.symm.surjective
nontriviality S
exfalso
exact not_nontrivial_iff_subsingleton.mpr ‹Subsingleton S› ‹Nontrivial S›
haveI : Inf... |
/-
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.Init.Algebra.Classes
import Mathlib.Data.FunLike.Basic
import Mathlib.Logic.Embedding.Basic
import Mathlib.Order.RelClasses
#align_import order.rel_is... | Mathlib/Order/RelIso/Basic.lean | 458 | 465 | theorem wellFounded_lift₂_iff [Setoid α] {r : α → α → Prop}
{H : ∀ (a₁ b₁ a₂ b₂ : α), a₁ ≈ a₂ → b₁ ≈ b₂ → r a₁ b₁ = r a₂ b₂} :
WellFounded (Quotient.lift₂ r H) ↔ WellFounded r := by |
constructor
· exact RelHomClass.wellFounded (Quotient.mkRelHom H)
· refine fun wf => ⟨fun q => ?_⟩
obtain ⟨a, rfl⟩ := q.exists_rep
exact acc_lift₂_iff.2 (wf.apply a)
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 649 | 651 | theorem coe_support_append {u v w : V} (p : G.Walk u v) (p' : G.Walk v w) :
((p.append p').support : Multiset V) = {u} + p.support.tail + p'.support.tail := by |
rw [support_append, ← Multiset.coe_add, coe_support]
|
/-
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.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 163 | 169 | theorem coeff_mul_of_natDegree_le (pm : p.natDegree ≤ m) (qn : q.natDegree ≤ n) :
(p * q).coeff (m + n) = p.coeff m * q.coeff n := by |
simp_rw [← Polynomial.toFinsupp_apply, toFinsupp_mul]
refine AddMonoidAlgebra.apply_add_of_supDegree_le ?_ Function.injective_id ?_ ?_
· simp
· rwa [supDegree_eq_natDegree, id_eq]
· rwa [supDegree_eq_natDegree, id_eq]
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 325 | 337 | theorem AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one [FiniteDimensional k V]
[Fintype ι] {p : ι → P} (hi : AffineIndependent k p) :
affineSpan k (Set.range p) = ⊤ ↔ Fintype.card ι = finrank k V + 1 := by |
constructor
· intro h_tot
let n := Fintype.card ι - 1
have hn : Fintype.card ι = n + 1 :=
(Nat.succ_pred_eq_of_pos (card_pos_of_affineSpan_eq_top k V P h_tot)).symm
rw [hn, ← finrank_top, ← (vectorSpan_eq_top_of_affineSpan_eq_top k V P) h_tot,
← hi.finrank_vectorSpan hn]
· intro hc
rw... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 203 | 205 | theorem restrict_comm (hs : MeasurableSet s) :
(μ.restrict t).restrict s = (μ.restrict s).restrict t := by |
rw [restrict_restrict hs, restrict_restrict' hs, inter_comm]
|
/-
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.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 522 | 532 | theorem strong_law_aux3 :
(fun n => 𝔼[∑ i ∈ range n, truncation (X i) i] - n * 𝔼[X 0]) =o[atTop] ((↑) : ℕ → ℝ) := by |
have A : Tendsto (fun i => 𝔼[truncation (X i) i]) atTop (𝓝 𝔼[X 0]) := by
convert (tendsto_integral_truncation hint).comp tendsto_natCast_atTop_atTop using 1
ext i
exact (hident i).truncation.integral_eq
convert Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 A) using 1
... |
/-
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.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 517 | 529 | theorem Dense.topology_eq_generateFrom [DenselyOrdered α] {s : Set α} (hs : Dense s) :
‹TopologicalSpace α› = .generateFrom (Ioi '' s ∪ Iio '' s) := by |
refine (OrderTopology.topology_eq_generate_intervals (α := α)).trans ?_
refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)
· simp only [union_subset_iff, image_subset_iff]
exact ⟨fun a _ ↦ ⟨a, .inl rfl⟩, fun a _ ↦ ⟨a, .inr rfl⟩⟩
· rintro _ ⟨a, rfl | rfl⟩
· rw [hs.Ioi_eq_biUnion]
let _... |
/-
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.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
/-!
# Perfect fields and rings
In this f... | Mathlib/FieldTheory/Perfect.lean | 308 | 310 | theorem roots_expand : (expand R p f).roots = p • f.roots.map (frobeniusEquiv R p).symm := by |
conv_lhs => rw [← pow_one p, roots_expand_pow, iterateFrobeniusEquiv_eq_pow, pow_one]
rfl
|
/-
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 | 177 | 191 | theorem exists_measurable_le_lintegral_eq (f : α → ℝ≥0∞) :
∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ ∫⁻ a, f a ∂μ = ∫⁻ a, g a ∂μ := by |
rcases eq_or_ne (∫⁻ a, f a ∂μ) 0 with h₀ | h₀
· exact ⟨0, measurable_zero, zero_le f, h₀.trans lintegral_zero.symm⟩
rcases exists_seq_strictMono_tendsto' h₀.bot_lt with ⟨L, _, hLf, hL_tendsto⟩
have : ∀ n, ∃ g : α → ℝ≥0∞, Measurable g ∧ g ≤ f ∧ L n < ∫⁻ a, g a ∂μ := by
intro n
simpa only [← iSup_lintegr... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 773 | 774 | theorem reflection_bot : reflection (⊥ : Submodule 𝕜 E) = LinearIsometryEquiv.neg 𝕜 := by |
ext; simp [reflection_apply]
|
/-
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.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 74 | 77 | theorem coeff_p [CharP R p] (i : ℕ) : (p : 𝕎 R).coeff i = if i = 1 then 1 else 0 := by |
split_ifs with hi
· simpa only [hi, pow_one] using coeff_p_pow p R 1
· simpa only [pow_one] using coeff_p_pow_eq_zero p R hi
|
/-
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
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 858 | 859 | theorem mk_quot_out [Monoid α] (a : α) : Quot.out (Associates.mk a) ~ᵤ a := by |
rw [← Associates.mk_eq_mk_iff_associated, Associates.quot_out]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f... | Mathlib/Data/PNat/Xgcd.lean | 506 | 516 | theorem gcd_eq : gcdD a b = gcd a b := by |
rcases gcd_props a b with ⟨_, h₁, h₂, _, _, h₅, _⟩
apply dvd_antisymm
· apply dvd_gcd
· exact Dvd.intro (gcdA' a b) (h₁.trans (mul_comm _ _)).symm
· exact Dvd.intro (gcdB' a b) (h₂.trans (mul_comm _ _)).symm
· have h₇ : (gcd a b : ℕ) ∣ gcdZ a b * a := (Nat.gcd_dvd_left a b).trans (dvd_mul_left _ _)
... |
/-
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.Analysis.NormedSpace.AffineIsometry
import Mathlib.Topology.Algebra.ContinuousAffineMap
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_... | Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | 66 | 67 | theorem coe_contLinear_eq_linear (f : P →ᴬ[R] Q) :
(f.contLinear : V →ₗ[R] W) = (f : P →ᵃ[R] Q).linear := by | ext; rfl
|
/-
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.OfAssociative
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
#align_import algebra.lie.matrix fro... | Mathlib/Algebra/Lie/Matrix.lean | 76 | 79 | theorem Matrix.lieConj_symm_apply (P A : Matrix n n R) (h : Invertible P) :
(P.lieConj h).symm A = P⁻¹ * A * P := by |
simp [LinearEquiv.symm_conj_apply, Matrix.lieConj, LinearMap.toMatrix'_comp,
LinearMap.toMatrix'_toLin']
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 183 | 186 | theorem lift_comp_ι (g : CliffordAlgebra Q →ₐ[R] A) :
lift Q ⟨g.toLinearMap.comp (ι Q), comp_ι_sq_scalar _⟩ = g := by |
-- Porting note: removed `rw [lift_symm_apply]; rfl`, changed `convert` to `exact`
exact (lift Q : _ ≃ (CliffordAlgebra Q →ₐ[R] A)).apply_symm_apply g
|
/-
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.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
/-!
# Fractional ideal norms
This file defines the absolute ideal norm of a frac... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 132 | 147 | theorem absNorm_span_singleton [Module.Finite ℚ K] (x : K) :
absNorm (spanSingleton R⁰ x) = |(Algebra.norm ℚ x)| := by |
have : IsDomain K := IsFractionRing.isDomain R
obtain ⟨d, ⟨r, hr⟩⟩ := IsLocalization.exists_integer_multiple R⁰ x
rw [absNorm_eq' d (Ideal.span {r})]
· rw [Ideal.absNorm_span_singleton]
simp_rw [Int.cast_natAbs, Int.cast_abs, show ((Algebra.norm ℤ _) : ℚ) = algebraMap ℤ ℚ
(Algebra.norm ℤ _) by rfl, ←... |
/-
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, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 235 | 236 | theorem coprod_inr (f : M →ₗ[R] M₃) (g : M₂ →ₗ[R] M₃) : (coprod f g).comp (inr R M M₂) = g := by |
ext; simp only [map_zero, coprod_apply, inr_apply, zero_add, comp_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, 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 | 1,674 | 1,675 | theorem aleph0_le_add_iff {a b : Cardinal} : ℵ₀ ≤ a + b ↔ ℵ₀ ≤ a ∨ ℵ₀ ≤ b := by |
simp only [← not_lt, add_lt_aleph0_iff, not_and_or]
|
/-
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, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 514 | 527 | theorem span_preimage_eq [RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {s : Set M₂} (h₀ : s.Nonempty)
(h₁ : s ⊆ range f) : span R (f ⁻¹' s) = (span R₂ s).comap f := by |
suffices (span R₂ s).comap f ≤ span R (f ⁻¹' s) by exact le_antisymm (span_preimage_le f s) this
have hk : ker f ≤ span R (f ⁻¹' s) := by
let y := Classical.choose h₀
have hy : y ∈ s := Classical.choose_spec h₀
rw [ker_le_iff]
use y, h₁ hy
rw [← Set.singleton_subset_iff] at hy
exact Set.Sub... |
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