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) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 401 | 403 | theorem pullbackP1Iso_hom_fst (i : 𝒰.J) :
(pullbackP1Iso 𝒰 f g i).hom ≫ pullback.fst = pullback.snd := by |
simp_rw [pullbackP1Iso, pullback.lift_fst]
|
/-
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.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 101 | 103 | theorem withDensityᵥ_sub (hf : Integrable f μ) (hg : Integrable g μ) :
μ.withDensityᵥ (f - g) = μ.withDensityᵥ f - μ.withDensityᵥ g := by |
rw [sub_eq_add_neg, sub_eq_add_neg, withDensityᵥ_add hf hg.neg, withDensityᵥ_neg]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Chris Hughes
-/
import Mathlib.Data.List.Nodup
#align_import data.list.duplicate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
/-!
... | Mathlib/Data/List/Duplicate.lean | 70 | 73 | theorem Duplicate.ne_singleton (h : x ∈+ l) (y : α) : l ≠ [y] := by |
induction' h with l' h z l' h _
· simp [ne_nil_of_mem h]
· simp [ne_nil_of_mem h.mem]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 301 | 309 | theorem toLin_restrict_range_dualCoannihilator_eq_orthogonal (B : BilinForm K V)
(W : Subspace K V) : (B.domRestrict W).range.dualCoannihilator = B.orthogonal W := by |
ext x; constructor <;> rw [mem_orthogonal_iff] <;> intro hx
· intro y hy
rw [Submodule.mem_dualCoannihilator] at hx
exact hx (B.domRestrict W ⟨y, hy⟩) ⟨⟨y, hy⟩, rfl⟩
· rw [Submodule.mem_dualCoannihilator]
rintro _ ⟨⟨w, hw⟩, rfl⟩
exact hx w hw
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Ker.lean | 187 | 187 | theorem sub_mem_ker_iff {x y} : x - y ∈ ker f ↔ f x = f y := by | rw [mem_ker, map_sub, sub_eq_zero]
|
/-
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.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.Layercake
#align_import measure_theory.integral.layercake from "leanprover-commu... | Mathlib/Analysis/SpecialFunctions/Pow/Integral.lean | 50 | 72 | theorem lintegral_rpow_eq_lintegral_meas_le_mul :
∫⁻ ω, ENNReal.ofReal (f ω ^ p) ∂μ =
ENNReal.ofReal p * ∫⁻ t in Ioi 0, μ {a : α | t ≤ f a} * ENNReal.ofReal (t ^ (p - 1)) := by |
have one_lt_p : -1 < p - 1 := by linarith
have obs : ∀ x : ℝ, ∫ t : ℝ in (0)..x, t ^ (p - 1) = x ^ p / p := by
intro x
rw [integral_rpow (Or.inl one_lt_p)]
simp [Real.zero_rpow p_pos.ne.symm]
set g := fun t : ℝ => t ^ (p - 1)
have g_nn : ∀ᵐ t ∂volume.restrict (Ioi (0 : ℝ)), 0 ≤ g t := by
filter... |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8... | Mathlib/LinearAlgebra/SModEq.lean | 44 | 44 | theorem sub_mem : x ≡ y [SMOD U] ↔ x - y ∈ U := by | rw [SModEq.def, Submodule.Quotient.eq]
|
/-
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.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 440 | 442 | theorem exists_lt_mem_ball_of_mem_ball (h : x ∈ ball y ε) : ∃ ε' < ε, x ∈ ball y ε' := by |
simp only [mem_ball] at h ⊢
exact ⟨(dist x y + ε) / 2, by linarith, by linarith⟩
|
/-
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 | 698 | 722 | theorem HasGroupoid.comp
(H : ∀ e ∈ G₂, LiftPropOn (IsLocalStructomorphWithinAt G₁) (e : H₂ → H₂) e.source) :
@HasGroupoid H₁ _ H₃ _ (ChartedSpace.comp H₁ H₂ H₃) G₁ :=
let _ := ChartedSpace.comp H₁ H₂ H₃ -- Porting note: need this to synthesize `ChartedSpace H₁ H₃`
{ compatible := by |
rintro _ _ ⟨e, he, f, hf, rfl⟩ ⟨e', he', f', hf', rfl⟩
apply G₁.locality
intro x hx
simp only [mfld_simps] at hx
have hxs : x ∈ f.symm ⁻¹' (e.symm ≫ₕ e').source := by simp only [hx, mfld_simps]
have hxs' : x ∈ f.target ∩ f.symm ⁻¹' ((e.symm ≫ₕ e').source ∩ e.symm ≫ₕ e' ⁻¹' f'.source... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 740 | 741 | theorem mul_lt_of_lt_one_left [MulPosStrictMono α] (hb : 0 < b) (h : a < 1) : a * b < b := by |
simpa only [one_mul] using mul_lt_mul_of_pos_right h hb
|
/-
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, Floris van Doorn, Yaël Dillies
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monoto... | Mathlib/Data/Nat/Factorial/Basic.lean | 121 | 129 | theorem factorial_inj (hn : 1 < n) : n ! = m ! ↔ n = m := by |
refine ⟨fun h => ?_, congr_arg _⟩
obtain hnm | rfl | hnm := lt_trichotomy n m
· rw [← factorial_lt <| lt_of_succ_lt hn, h] at hnm
cases lt_irrefl _ hnm
· rfl
rw [← one_lt_factorial, h, one_lt_factorial] at hn
rw [← factorial_lt <| lt_of_succ_lt hn, h] at hnm
cases lt_irrefl _ hnm
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 586 | 601 | theorem infinite_not_isOfFinOrder {x : G} (h : ¬IsOfFinOrder x) :
{ y : G | ¬IsOfFinOrder y }.Infinite := by |
let s := { n | 0 < n }.image fun n : ℕ => x ^ n
have hs : s ⊆ { y : G | ¬IsOfFinOrder y } := by
rintro - ⟨n, hn : 0 < n, rfl⟩ (contra : IsOfFinOrder (x ^ n))
apply h
rw [isOfFinOrder_iff_pow_eq_one] at contra ⊢
obtain ⟨m, hm, hm'⟩ := contra
exact ⟨n * m, mul_pos hn hm, by rwa [pow_mul]⟩
suffi... |
/-
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 | 1,782 | 1,785 | theorem binary_relation_sInf_iff {α β : Type*} (s : Set (α → β → Prop)) {a : α} {b : β} :
sInf s a b ↔ ∀ r : α → β → Prop, r ∈ s → r a b := by |
rw [sInf_apply]
simp [← eq_iff_iff]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 386 | 394 | theorem HasLineDerivAt.le_of_lip' {f : E → F} {f' : F} {x₀ : E} (hf : HasLineDerivAt 𝕜 f f' x₀ v)
{C : ℝ} (hC₀ : 0 ≤ C) (hlip : ∀ᶠ x in 𝓝 x₀, ‖f x - f x₀‖ ≤ C * ‖x - x₀‖) :
‖f'‖ ≤ C * ‖v‖ := by |
apply HasDerivAt.le_of_lip' hf (by positivity)
have A : Continuous (fun (t : 𝕜) ↦ x₀ + t • v) := by continuity
have : ∀ᶠ x in 𝓝 (x₀ + (0 : 𝕜) • v), ‖f x - f x₀‖ ≤ C * ‖x - x₀‖ := by simpa using hlip
filter_upwards [(A.continuousAt (x := 0)).preimage_mem_nhds this] with t ht
simp only [preimage_setOf_eq, a... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 2,118 | 2,121 | theorem iSup_insert (a : α) (s : Finset α) (t : α → β) :
⨆ x ∈ insert a s, t x = t a ⊔ ⨆ x ∈ s, t x := by |
rw [insert_eq]
simp only [iSup_union, Finset.iSup_singleton]
|
/-
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
-/
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Mea... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 465 | 468 | theorem ae_measure_lt_top {s : Set (α × β)} (hs : MeasurableSet s) (h2s : (μ.prod ν) s ≠ ∞) :
∀ᵐ x ∂μ, ν (Prod.mk x ⁻¹' s) < ∞ := by |
rw [prod_apply hs] at h2s
exact ae_lt_top (measurable_measure_prod_mk_left hs) h2s
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 62 | 64 | theorem support_scaleRoots_le (p : R[X]) (s : R) : (scaleRoots p s).support ≤ p.support := by |
intro
simpa using left_ne_zero_of_mul
|
/-
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, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 581 | 585 | theorem padicValNat_dvd_iff (n : ℕ) [hp : Fact p.Prime] (a : ℕ) :
p ^ n ∣ a ↔ a = 0 ∨ n ≤ padicValNat p a := by |
rcases eq_or_ne a 0 with (rfl | ha)
· exact iff_of_true (dvd_zero _) (Or.inl rfl)
· rw [padicValNat_dvd_iff_le ha, or_iff_right ha]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,198 | 1,202 | theorem dualAnnihilator_dualAnnihilator_eq (W : Subspace K V) :
W.dualAnnihilator.dualAnnihilator = Module.mapEvalEquiv K V W := by |
have : _ = W := Subspace.dualAnnihilator_dualCoannihilator_eq
rw [dualCoannihilator, ← Module.mapEvalEquiv_symm_apply] at this
rwa [← OrderIso.symm_apply_eq]
|
/-
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 | 330 | 333 | theorem nhds_top_basis [TopologicalSpace α] [LinearOrder α] [OrderTop α] [OrderTopology α]
[Nontrivial α] : (𝓝 ⊤).HasBasis (fun a : α => a < ⊤) fun a : α => Ioi a := by |
have : ∃ x : α, x < ⊤ := (exists_ne ⊤).imp fun x hx => hx.lt_top
simpa only [Iic_top, nhdsWithin_univ, Ioc_top] using nhdsWithin_Iic_basis' this
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulA... | Mathlib/Topology/Algebra/Module/Basic.lean | 818 | 820 | theorem comp_zero (g : M₂ →SL[σ₂₃] M₃) : g.comp (0 : M₁ →SL[σ₁₂] M₂) = 0 := by |
ext
simp
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 467 | 470 | theorem IsDiag.mem_range_diag {z : Sym2 α} : IsDiag z → z ∈ Set.range (@diag α) := by |
induction' z using Sym2.ind with x y
rintro (rfl : x = y)
exact ⟨_, rfl⟩
|
/-
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.RBMap.Alter
import Batteries.Data.List.Lemmas
/-!
# Additional lemmas for Red-black trees
-/
namespace Batteries
namespace RBNode
open RBColor... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 241 | 244 | theorem min?_mem {t : RBNode α} (h : t.min? = some a) : a ∈ t := by |
rw [min?_eq_toList_head?] at h
rw [← mem_toList]
revert h; cases toList t <;> rintro ⟨⟩; constructor
|
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Interval/Set/Group.lean | 226 | 228 | theorem pairwise_disjoint_Ioo_zpow :
Pairwise (Disjoint on fun n : ℤ => Ioo (b ^ n) (b ^ (n + 1))) := by |
simpa only [one_mul] using pairwise_disjoint_Ioo_mul_zpow 1 b
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,255 | 1,266 | theorem map_comp_map {A : Type*} [CommMonoid A] {U : Submonoid A} {R} [CommMonoid R]
(j : LocalizationMap U R) {l : P →* A} (hl : ∀ w : T, l w ∈ U) :
(k.map hl j).comp (f.map hy k) =
f.map (fun x ↦ show l.comp g x ∈ U from hl ⟨g x, hy x⟩) j := by |
ext z
show j.toMap _ * _ = j.toMap (l _) * _
rw [mul_inv_left, ← mul_assoc, mul_inv_right]
show j.toMap _ * j.toMap (l (g _)) = j.toMap (l _) * _
rw [← j.toMap.map_mul, ← j.toMap.map_mul, ← l.map_mul, ← l.map_mul]
exact
k.comp_eq_of_eq hl j
(by rw [k.toMap.map_mul, k.toMap.map_mul, sec_spec', mul... |
/-
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 | 2,565 | 2,568 | theorem set_walk_length_toFinset_eq (n : ℕ) (u v : V) :
{p : G.Walk u v | p.length = n}.toFinset = G.finsetWalkLength n u v := by |
ext p
simp [← coe_finsetWalkLength_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.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76... | Mathlib/RingTheory/IsTensorProduct.lean | 249 | 280 | theorem IsBaseChange.of_lift_unique
(h : ∀ (Q : Type max v₁ v₂ v₃) [AddCommMonoid Q],
∀ [Module R Q] [Module S Q], ∀ [IsScalarTower R S Q],
∀ g : M →ₗ[R] Q, ∃! g' : N →ₗ[S] Q, (g'.restrictScalars R).comp f = g) :
IsBaseChange S f := by |
obtain ⟨g, hg, -⟩ :=
h (ULift.{v₂} <| S ⊗[R] M)
(ULift.moduleEquiv.symm.toLinearMap.comp <| TensorProduct.mk R S M 1)
let f' : S ⊗[R] M →ₗ[R] N :=
TensorProduct.lift (((LinearMap.flip (AlgHom.toLinearMap (Algebra.ofId S
(Module.End S (M →ₗ[R] N))))) f).restrictScalars R)
change Function.Bijec... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,468 | 1,471 | theorem factors_mk (a : α) (h : a ≠ 0) : (Associates.mk a).factors = factors' a := by |
classical
apply dif_neg
apply mt mk_eq_zero.1 h
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
#align_import algebraic_geomet... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 599 | 604 | theorem range_pullback_to_base_of_right :
Set.range (pullback.fst ≫ g : pullback g f ⟶ Z).1.base =
Set.range g.1.base ∩ Set.range f.1.base := by |
rw [Scheme.comp_val_base, TopCat.coe_comp, Set.range_comp, range_pullback_fst_of_right,
Opens.map_obj, Opens.carrier_eq_coe, Opens.coe_mk, Set.image_preimage_eq_inter_range,
Set.inter_comm]
|
/-
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 | 393 | 394 | theorem mkQ_map_self : map p.mkQ p = ⊥ := by |
rw [eq_bot_iff, map_le_iff_le_comap, comap_bot, ker_mkQ]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,159 | 1,175 | theorem contDiffOn_succ_iff_fderivWithin {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1 : ℕ) f s ↔
DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 n (fun y => fderivWithin 𝕜 f s y) s := by |
refine ⟨fun H => ?_, fun h => contDiffOn_succ_of_fderivWithin h.1 h.2⟩
refine ⟨H.differentiableOn (WithTop.coe_le_coe.2 (Nat.le_add_left 1 n)), fun x hx => ?_⟩
rcases contDiffWithinAt_succ_iff_hasFDerivWithinAt.1 (H x hx) with ⟨u, hu, f', hff', hf'⟩
rcases mem_nhdsWithin.1 hu with ⟨o, o_open, xo, ho⟩
rw [int... |
/-
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 | 225 | 227 | theorem isClosed_iff_of_mem (h : ∞ ∈ s) : IsClosed s ↔ IsClosed ((↑) ⁻¹' s : Set X) := by |
have : ∞ ∉ sᶜ := fun H => H h
rw [← isOpen_compl_iff, isOpen_iff_of_not_mem this, ← isOpen_compl_iff, preimage_compl]
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 511 | 515 | theorem diagonal_map [Zero α] [Zero β] {f : α → β} (h : f 0 = 0) {d : n → α} :
(diagonal d).map f = diagonal fun m => f (d m) := by |
ext
simp only [diagonal_apply, map_apply]
split_ifs <;> simp [h]
|
/-
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.Basic
import Mathlib.Init.Order.Defs
import Mathlib.Order.Monotone.Basic
import Mathlib.Order.ULift
import Mathlib.Tactic.GCongr.Core
#align... | Mathlib/Order/Lattice.lean | 799 | 804 | theorem le_sup_iff : a ≤ b ⊔ c ↔ a ≤ b ∨ a ≤ c := by |
exact ⟨fun h =>
(le_total c b).imp
(fun bc => by rwa [sup_eq_left.2 bc] at h)
(fun bc => by rwa [sup_eq_right.2 bc] at h),
fun h => h.elim le_sup_of_le_left le_sup_of_le_right⟩
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 146 | 151 | theorem aecover_closedBall {x : α} {r : ι → ℝ} (hr : Tendsto r l atTop) :
AECover μ l (fun i ↦ Metric.closedBall x (r i)) where
measurableSet _ := Metric.isClosed_ball.measurableSet
ae_eventually_mem := by |
filter_upwards with y
filter_upwards [hr (Ici_mem_atTop (dist x y))] with a ha using by simpa [dist_comm] using ha
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.... | Mathlib/Analysis/Convex/Topology.lean | 296 | 301 | theorem Set.Finite.isCompact_convexHull {s : Set E} (hs : s.Finite) :
IsCompact (convexHull ℝ s) := by |
rw [hs.convexHull_eq_image]
apply (@isCompact_stdSimplex _ hs.fintype).image
haveI := hs.fintype
apply LinearMap.continuous_on_pi
|
/-
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 | 312 | 315 | theorem oangle_sub_left_eq_arctan_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x - y) x = Real.arctan (‖y‖ / ‖x‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
exact (-o).oangle_sub_right_eq_arctan_of_oangle_eq_pi_div_two h
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization... | Mathlib/RingTheory/LocalProperties.lean | 484 | 504 | theorem multiple_mem_span_of_mem_localization_span [Algebra R' S] [Algebra R S]
[IsScalarTower R R' S] [IsLocalization M R'] (s : Set S) (x : S)
(hx : x ∈ Submodule.span R' s) : ∃ t : M, t • x ∈ Submodule.span R s := by |
obtain ⟨s', hss', hs'⟩ := Submodule.mem_span_finite_of_mem_span hx
rsuffices ⟨t, ht⟩ : ∃ t : M, t • x ∈ Submodule.span R (s' : Set S)
· exact ⟨t, Submodule.span_mono hss' ht⟩
clear hx hss' s
induction s' using Finset.induction_on generalizing x
· use 1; simpa using hs'
rename_i a s _ hs
simp only [Fins... |
/-
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.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform... | Mathlib/Topology/UniformSpace/Cauchy.lean | 249 | 253 | theorem CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n := by |
rw [cauchySeq_iff_tendsto] at hu
exact ((hu.comp <| hf.prod_atTop hg).comp tendsto_atTop_diagonal).subseq_mem hV
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 708 | 709 | theorem prod_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(∏ i ∈ s, J i) ⊔ I = ⊤ := by | rw [sup_comm, sup_prod_eq_top]; intro i hi; rw [sup_comm, h i hi]
|
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 68 | 69 | theorem catalan_succ (n : ℕ) : catalan (n + 1) = ∑ i : Fin n.succ, catalan i * catalan (n - i) := by |
rw [catalan]
|
/-
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 | 495 | 500 | theorem sum_roots_eq_nextCoeff_of_monic_of_split {P : K[X]} (hmo : P.Monic)
(hP : P.Splits (RingHom.id K)) : P.nextCoeff = -P.roots.sum := by |
nth_rw 1 [eq_prod_roots_of_monic_of_splits_id hmo hP]
rw [Monic.nextCoeff_multiset_prod _ _ fun a ha => _]
· simp_rw [nextCoeff_X_sub_C, Multiset.sum_map_neg']
· simp only [monic_X_sub_C, implies_true]
|
/-
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.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 301 | 305 | theorem IsClosed.exists_closed_singleton [T0Space X] [CompactSpace X] {S : Set X}
(hS : IsClosed S) (hne : S.Nonempty) : ∃ x : X, x ∈ S ∧ IsClosed ({x} : Set X) := by |
obtain ⟨V, Vsub, Vne, Vcls, hV⟩ := hS.exists_minimal_nonempty_closed_subset hne
rcases minimal_nonempty_closed_eq_singleton Vcls Vne hV with ⟨x, rfl⟩
exact ⟨x, Vsub (mem_singleton x), Vcls⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,015 | 1,031 | theorem prod_eq_mul_of_mem {s : Finset α} {f : α → β} (a b : α) (ha : a ∈ s) (hb : b ∈ s)
(hn : a ≠ b) (h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) : ∏ x ∈ s, f x = f a * f b := by |
haveI := Classical.decEq α; let s' := ({a, b} : Finset α)
have hu : s' ⊆ s := by
refine insert_subset_iff.mpr ?_
apply And.intro ha
apply singleton_subset_iff.mpr hb
have hf : ∀ c ∈ s, c ∉ s' → f c = 1 := by
intro c hc hcs
apply h₀ c hc
apply not_or.mp
intro hab
apply hcs
rw [... |
/-
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.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import proba... | Mathlib/Probability/Density.lean | 168 | 170 | theorem measurable_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
(μ : Measure E := by | volume_tac) : Measurable (pdf X ℙ μ) := by
exact measurable_rnDeriv _ _
|
/-
Copyright (c) 2023 Xavier Généreux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Généreux, Patrick Massot
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Analysis.RCLike.Basic
/-!
# A collection of specific limit computations for `RCLike`
-... | Mathlib/Analysis/SpecificLimits/RCLike.lean | 19 | 22 | theorem RCLike.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : ℕ => (n : 𝕜)⁻¹) atTop (𝓝 0) := by |
convert tendsto_algebraMap_inverse_atTop_nhds_zero_nat 𝕜
simp
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 52 | 59 | theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
Subobject.Factors (NormalizedMooreComplex.objX X n) (PInfty.f n) := by |
rcases n with _|n
· apply top_factors
· rw [PInfty_f, NormalizedMooreComplex.objX, finset_inf_factors]
intro i _
apply kernelSubobject_factors
exact (HigherFacesVanish.of_P (n + 1) n) i le_add_self
|
/-
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, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da... | Mathlib/Algebra/Group/Ext.lean | 71 | 74 | theorem LeftCancelMonoid.toMonoid_injective {M : Type u} :
Function.Injective (@LeftCancelMonoid.toMonoid M) := by |
rintro @⟨@⟨⟩⟩ @⟨@⟨⟩⟩ h
congr <;> injection h
|
/-
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 | 586 | 597 | theorem isCoseparator_prod (G H : C) [HasBinaryProduct G H] :
IsCoseparator (G ⨯ H) ↔ IsCoseparating ({G, H} : Set C) := by |
refine
⟨fun h X Y u v huv => ?_, fun h =>
(isCoseparator_def _).2 fun X Y u v huv => h _ _ fun Z hZ g => ?_⟩
· refine h.def _ _ fun g => prod.hom_ext ?_ ?_
· simpa using huv G (by simp) (g ≫ Limits.prod.fst)
· simpa using huv H (by simp) (g ≫ Limits.prod.snd)
· simp only [Set.mem_insert_iff, Se... |
/-
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.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
/-!
# Exact sequences with free modules
This file proves resu... | Mathlib/Algebra/Category/ModuleCat/Free.lean | 44 | 49 | theorem disjoint_span_sum : Disjoint (span R (range (u ∘ Sum.inl)))
(span R (range (u ∘ Sum.inr))) := by |
rw [huv, disjoint_comm]
refine Disjoint.mono_right (span_mono (range_comp_subset_range _ _)) ?_
rw [← LinearMap.range_coe, span_eq (LinearMap.range S.f), hS.moduleCat_range_eq_ker]
exact range_ker_disjoint hw
|
/-
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.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Commute.Hom
import Mathlib.Data.Fintype.Card
#align_import data.finset.noncomm_prod f... | Mathlib/Data/Finset/NoncommProd.lean | 329 | 333 | theorem noncommProd_insert_of_not_mem [DecidableEq α] (s : Finset α) (a : α) (f : α → β) (comm)
(ha : a ∉ s) :
noncommProd (insert a s) f comm =
f a * noncommProd s f (comm.mono fun _ => mem_insert_of_mem) := by |
simp only [← cons_eq_insert _ _ ha, noncommProd_cons]
|
/-
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 | 604 | 613 | theorem exists_pos_tsum_mul_lt_of_countable {ε : ℝ≥0∞} (hε : ε ≠ 0) {ι} [Countable ι] (w : ι → ℝ≥0∞)
(hw : ∀ i, w i ≠ ∞) : ∃ δ : ι → ℝ≥0, (∀ i, 0 < δ i) ∧ (∑' i, (w i * δ i : ℝ≥0∞)) < ε := by |
lift w to ι → ℝ≥0 using hw
rcases exists_pos_sum_of_countable hε ι with ⟨δ', Hpos, Hsum⟩
have : ∀ i, 0 < max 1 (w i) := fun i ↦ zero_lt_one.trans_le (le_max_left _ _)
refine ⟨fun i ↦ δ' i / max 1 (w i), fun i ↦ div_pos (Hpos _) (this i), ?_⟩
refine lt_of_le_of_lt (ENNReal.tsum_le_tsum fun i ↦ ?_) Hsum
rw [... |
/-
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 | 237 | 238 | theorem lapply_comp_lsingle_of_ne (a a' : α) (h : a ≠ a') :
lapply a ∘ₗ lsingle a' = (0 : M →ₗ[R] M) := by | ext; simp [h.symm]
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Ker.lean | 125 | 126 | theorem ker_codRestrict {τ₂₁ : R₂ →+* R} (p : Submodule R M) (f : M₂ →ₛₗ[τ₂₁] M) (hf) :
ker (codRestrict p f hf) = ker f := by | rw [ker, comap_codRestrict, Submodule.map_bot]; rfl
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison, Apurva Nakade
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.SetTheory.Game.PGame
import Mathlib.Tactic.Abel
#align_... | Mathlib/SetTheory/Game/Basic.lean | 111 | 113 | theorem not_le : ∀ {x y : Game}, ¬x ≤ y ↔ y ⧏ x := by |
rintro ⟨x⟩ ⟨y⟩
exact PGame.not_le
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Antoine Chambert-Loir
-/
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_impo... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 361 | 365 | theorem finsuppTensorFinsupp'_symm_single_eq_tmul_single_one (i : ι × κ) (r : R) :
(finsuppTensorFinsupp' R ι κ).symm (Finsupp.single i r) =
Finsupp.single i.1 r ⊗ₜ Finsupp.single i.2 1 := by |
nth_rw 1 [← mul_one r]
exact finsuppTensorFinsupp'_symm_single_mul R ι κ i _ _
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 750 | 758 | theorem eq_bot_or_top : I = ⊥ ∨ I = ⊤ := by |
rw [or_iff_not_imp_right]
change _ ≠ _ → _
rw [Ideal.ne_top_iff_one]
intro h1
rw [eq_bot_iff]
intro r hr
by_cases H : r = 0; · simpa
simpa [H, h1] using I.mul_mem_left r⁻¹ hr
|
/-
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 | 110 | 112 | theorem kernelSubobject_arrow_comp : (kernelSubobject f).arrow ≫ f = 0 := by |
rw [← kernelSubobject_arrow]
simp only [Category.assoc, kernel.condition, comp_zero]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
impor... | Mathlib/LinearAlgebra/Vandermonde.lean | 142 | 150 | theorem det_vandermonde_eq_zero_iff [IsDomain R] {n : ℕ} {v : Fin n → R} :
det (vandermonde v) = 0 ↔ ∃ i j : Fin n, v i = v j ∧ i ≠ j := by |
constructor
· simp only [det_vandermonde v, Finset.prod_eq_zero_iff, sub_eq_zero, forall_exists_index]
rintro i ⟨_, j, h₁, h₂⟩
exact ⟨j, i, h₂, (mem_Ioi.mp h₁).ne'⟩
· simp only [Ne, forall_exists_index, and_imp]
refine fun i j h₁ h₂ => Matrix.det_zero_of_row_eq h₂ (funext fun k => ?_)
rw [vanderm... |
/-
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.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 218 | 223 | theorem hasStrictFDerivAt_norm_sq (x : F) :
HasStrictFDerivAt (fun x => ‖x‖ ^ 2) (2 • (innerSL ℝ x)) x := by |
simp only [sq, ← @inner_self_eq_norm_mul_norm ℝ]
convert (hasStrictFDerivAt_id x).inner ℝ (hasStrictFDerivAt_id x)
ext y
simp [two_smul, real_inner_comm]
|
/-
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 | 967 | 971 | theorem lintegral_sub' {f g : α → ℝ≥0∞} (hg : AEMeasurable g μ) (hg_fin : ∫⁻ a, g a ∂μ ≠ ∞)
(h_le : g ≤ᵐ[μ] f) : ∫⁻ a, f a - g a ∂μ = ∫⁻ a, f a ∂μ - ∫⁻ a, g a ∂μ := by |
refine ENNReal.eq_sub_of_add_eq hg_fin ?_
rw [← lintegral_add_right' _ hg]
exact lintegral_congr_ae (h_le.mono fun x hx => tsub_add_cancel_of_le hx)
|
/-
Copyright (c) 2023 Andrew Yang, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Mi... | Mathlib/FieldTheory/KummerExtension.lean | 134 | 149 | theorem pow_ne_of_irreducible_X_pow_sub_C {n : ℕ} {a : K}
(H : Irreducible (X ^ n - C a)) {m : ℕ} (hm : m ∣ n) (hm' : m ≠ 1) (b : K) : b ^ m ≠ a := by |
have hn : n ≠ 0 := fun e ↦ not_irreducible_C
(1 - a) (by simpa only [e, pow_zero, ← C.map_one, ← map_sub] using H)
obtain ⟨k, rfl⟩ := hm
rintro rfl
obtain ⟨q, hq⟩ := sub_dvd_pow_sub_pow (X ^ k) (C b) m
rw [mul_comm, pow_mul, map_pow, hq] at H
have : degree q = 0 := by
simpa [isUnit_iff_degree_eq_ze... |
/-
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.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 120 | 126 | theorem mfderivWithin_eq_fderivWithin :
mfderivWithin 𝓘(𝕜, E) 𝓘(𝕜, E') f s x = fderivWithin 𝕜 f s x := by |
by_cases h : MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x
· simp only [mfderivWithin, h, if_pos, mfld_simps]
· simp only [mfderivWithin, h, if_neg, not_false_iff]
rw [mdifferentiableWithinAt_iff_differentiableWithinAt] at h
exact (fderivWithin_zero_of_not_differentiableWithinAt h).symm
|
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.alg... | Mathlib/Topology/Algebra/InfiniteSum/Module.lean | 167 | 178 | theorem ContinuousLinearEquiv.tsum_eq_iff [T2Space M] [T2Space M₂] {f : ι → M} (e : M ≃SL[σ] M₂)
{y : M₂} : (∑' z, e (f z)) = y ↔ ∑' z, f z = e.symm y := by |
by_cases hf : Summable f
· exact
⟨fun h ↦ (e.hasSum.mp ((e.summable.mpr hf).hasSum_iff.mpr h)).tsum_eq, fun h ↦
(e.hasSum.mpr (hf.hasSum_iff.mpr h)).tsum_eq⟩
· have hf' : ¬Summable fun z ↦ e (f z) := fun h ↦ hf (e.summable.mp h)
rw [tsum_eq_zero_of_not_summable hf, tsum_eq_zero_of_not_summable ... |
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 379 | 383 | theorem IsOrtho.map (f : E →ₗᵢ[𝕜] F) {U V : Submodule 𝕜 E} (h : U ⟂ V) : U.map f ⟂ V.map f := by |
rw [isOrtho_iff_inner_eq] at *
simp_rw [mem_map, forall_exists_index, and_imp, forall_apply_eq_imp_iff₂,
LinearIsometry.inner_map_map]
exact h
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 252 | 255 | theorem comp_tail {α : Type*} {β : Type*} (g : α → β) (q : Fin n.succ → α) :
g ∘ tail q = tail (g ∘ q) := by |
ext j
simp [tail]
|
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 303 | 304 | theorem minimalPeriod_pos_of_mem_periodicPts (hx : x ∈ periodicPts f) : 0 < minimalPeriod f x := by |
simp only [minimalPeriod, dif_pos hx, (Nat.find_spec hx).1.lt]
|
/-
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.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Co... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 490 | 496 | theorem factorial_smul (n : ℕ) :
n ! • p n (fun _ ↦ y) = iteratedFDeriv 𝕜 n f x (fun _ ↦ y) := by |
cases n
· rw [factorial_zero, one_smul, h.iteratedFDeriv_zero_apply_diag]
· rw [factorial_succ, mul_comm, mul_smul, ← derivSeries_apply_diag, ← smul_apply,
factorial_smul'.{_,u,v} _ h.fderiv, iteratedFDeriv_succ_apply_right]
rfl
|
/-
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.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import ... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 559 | 570 | theorem exists_smooth_zero_one_of_isClosed [T2Space M] [SigmaCompactSpace M] {s t : Set M}
(hs : IsClosed s) (ht : IsClosed t) (hd : Disjoint s t) :
∃ f : C^∞⟮I, M; 𝓘(ℝ), ℝ⟯, EqOn f 0 s ∧ EqOn f 1 t ∧ ∀ x, f x ∈ Icc 0 1 := by |
have : ∀ x ∈ t, sᶜ ∈ 𝓝 x := fun x hx => hs.isOpen_compl.mem_nhds (disjoint_right.1 hd hx)
rcases SmoothBumpCovering.exists_isSubordinate I ht this with ⟨ι, f, hf⟩
set g := f.toSmoothPartitionOfUnity
refine
⟨⟨_, g.smooth_sum⟩, fun x hx => ?_, fun x => g.sum_eq_one, fun x =>
⟨g.sum_nonneg x, g.sum_le_... |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 557 | 559 | theorem restrict_apply' (κ : kernel α β) (hs : MeasurableSet s) (a : α) (ht : MeasurableSet t) :
kernel.restrict κ hs a t = (κ a) (t ∩ s) := by |
rw [restrict_apply κ hs a, Measure.restrict_apply ht]
|
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 78 | 79 | theorem image_Iio (e : α ≃o β) (a : α) : e '' Iio a = Iio (e a) := by |
rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 118 | 127 | theorem isAcyclic_of_path_unique (h : ∀ (v w : V) (p q : G.Path v w), p = q) : G.IsAcyclic := by |
intro v c hc
simp only [Walk.isCycle_def, Ne] at hc
cases c with
| nil => cases hc.2.1 rfl
| cons ha c' =>
simp only [Walk.cons_isTrail_iff, Walk.support_cons, List.tail_cons, true_and_iff] at hc
specialize h _ _ ⟨c', by simp only [Walk.isPath_def, hc.2]⟩ (Path.singleton ha.symm)
rw [Path.singlet... |
/-
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.Algebra.Order.Nonneg.Ring
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Int.Lemmas
#align_import data.rat.nnrat fr... | Mathlib/Data/NNRat/Defs.lean | 382 | 387 | theorem ext_num_den (hn : p.num = q.num) (hd : p.den = q.den) : p = q := by |
refine ext <| Rat.ext ?_ ?_
· apply (Int.natAbs_inj_of_nonneg_of_nonneg _ _).1 hn
· exact Rat.num_nonneg.2 p.2
· exact Rat.num_nonneg.2 q.2
· exact hd
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 93 | 106 | theorem discr_zero_of_not_linearIndependent [IsDomain A] {b : ι → B}
(hli : ¬LinearIndependent A b) : discr A b = 0 := by |
classical
obtain ⟨g, hg, i, hi⟩ := Fintype.not_linearIndependent_iff.1 hli
have : (traceMatrix A b) *ᵥ g = 0 := by
ext i
have : ∀ j, (trace A B) (b i * b j) * g j = (trace A B) (g j • b j * b i) := by
intro j;
simp [mul_comm]
simp only [mulVec, dotProduct, traceMatrix_apply, Pi.zero_apply... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 764 | 765 | theorem length_toList (s : Multiset α) : s.toList.length = card s := by |
rw [← coe_card, coe_toList]
|
/-
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 | 755 | 769 | theorem ncard_congr {t : Set β} (f : ∀ a ∈ s, β) (h₁ : ∀ a ha, f a ha ∈ t)
(h₂ : ∀ a b ha hb, f a ha = f b hb → a = b) (h₃ : ∀ b ∈ t, ∃ a ha, f a ha = b) :
s.ncard = t.ncard := by |
set f' : s → t := fun x ↦ ⟨f x.1 x.2, h₁ _ _⟩
have hbij : f'.Bijective := by
constructor
· rintro ⟨x, hx⟩ ⟨y, hy⟩ hxy
simp only [f', Subtype.mk.injEq] at hxy ⊢
exact h₂ _ _ hx hy hxy
rintro ⟨y, hy⟩
obtain ⟨a, ha, rfl⟩ := h₃ y hy
simp only [Subtype.mk.injEq, Subtype.exists]
exact... |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 735 | 737 | theorem mem_support_iff : n ∈ p.support ↔ p.coeff n ≠ 0 := by |
rcases p with ⟨⟩
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 | 1,280 | 1,286 | theorem Submodule.exists_finset_of_mem_iSup {ι : Sort _} (p : ι → Submodule R M) {m : M}
(hm : m ∈ ⨆ i, p i) : ∃ s : Finset ι, m ∈ ⨆ i ∈ s, p i := by |
have :=
CompleteLattice.IsCompactElement.exists_finset_of_le_iSup (Submodule R M)
(Submodule.singleton_span_isCompactElement m) p
simp only [Submodule.span_singleton_le_iff_mem] at this
exact this hm
|
/-
Copyright (c) 2023 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.Analytic.Basic
/-! We specialize the theory fo analytic functions to the case of functions that admit a
development given by a *finite* formal mu... | Mathlib/Analysis/Analytic/CPolynomial.lean | 284 | 294 | theorem HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero (hf : ∀ y ∈ EMetric.ball x r, f y = 0)
(r_pos : 0 < r) (hp : ∀ n, p n = 0) : HasFiniteFPowerSeriesOnBall f p x 0 r := by |
refine ⟨⟨?_, r_pos, ?_⟩, fun n _ ↦ hp n⟩
· rw [p.radius_eq_top_of_forall_image_add_eq_zero 0 (fun n ↦ by rw [add_zero]; exact hp n)]
exact le_top
· intro y hy
rw [hf (x + y)]
· convert hasSum_zero
rw [hp, ContinuousMultilinearMap.zero_apply]
· rwa [EMetric.mem_ball, edist_eq_coe_nnnorm_sub,... |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | Mathlib/Order/Heyting/Regular.lean | 70 | 71 | theorem IsRegular.himp (ha : IsRegular a) (hb : IsRegular b) : IsRegular (a ⇨ b) := by |
rw [IsRegular, compl_compl_himp_distrib, ha.eq, hb.eq]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 233 | 242 | theorem prod_induction_nonempty (p : α → Prop) (p_mul : ∀ a b, p a → p b → p (a * b)) (hs : s ≠ ∅)
(p_s : ∀ a ∈ s, p a) : p s.prod := by |
-- Porting note: used to be `refine' Multiset.induction _ _`
induction' s using Multiset.induction_on with a s hsa
· simp at hs
rw [prod_cons]
by_cases hs_empty : s = ∅
· simp [hs_empty, p_s a]
have hps : ∀ x, x ∈ s → p x := fun x hxs => p_s x (mem_cons_of_mem hxs)
exact p_mul a s.prod (p_s a (mem_cons... |
/-
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 | 166 | 170 | theorem Filter.HasBasis.uniformEmbedding_iff {ι ι'} {p : ι → Prop} {p' : ι' → Prop} {s s'}
(h : (𝓤 α).HasBasis p s) (h' : (𝓤 β).HasBasis p' s') {f : α → β} :
UniformEmbedding f ↔ Injective f ∧ UniformContinuous f ∧
(∀ j, p j → ∃ i, p' i ∧ ∀ x y, (f x, f y) ∈ s' i → (x, y) ∈ s j) := by |
simp only [h.uniformEmbedding_iff' h', h.uniformContinuous_iff h']
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 84 | 91 | theorem continuousAt_cpow {p : ℂ × ℂ} (hp_fst : p.fst ∈ slitPlane) :
ContinuousAt (fun x : ℂ × ℂ => x.1 ^ x.2) p := by |
rw [continuousAt_congr (cpow_eq_nhds' <| slitPlane_ne_zero hp_fst)]
refine continuous_exp.continuousAt.comp ?_
exact
ContinuousAt.mul
(ContinuousAt.comp (continuousAt_clog hp_fst) continuous_fst.continuousAt)
continuous_snd.continuousAt
|
/-
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 | 1,008 | 1,010 | theorem isClosed_range_inr : IsClosed (range (inr : Y → X ⊕ Y)) := by |
rw [← isOpen_compl_iff, compl_range_inr]
exact isOpen_range_inl
|
/-
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 | 682 | 686 | theorem tan_oangle_left_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
Real.Angle.tan (∡ p₃ p₁ p₂) = dist p₃ p₂ / dist p₁ p₂ := by |
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.tan_coe,
tan_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.UniformSpace.Completion
#align_... | Mathlib/Topology/Algebra/GroupCompletion.lean | 278 | 285 | theorem AddMonoidHom.completion_zero :
AddMonoidHom.completion (0 : α →+ β) continuous_const = 0 := by |
ext x
refine Completion.induction_on x ?_ ?_
· apply isClosed_eq (AddMonoidHom.continuous_completion (0 : α →+ β) continuous_const)
simp [continuous_const]
· intro a
simp [(0 : α →+ β).completion_coe continuous_const, coe_zero]
|
/-
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 | 86 | 92 | theorem toMatrix_update [DecidableEq ι'] (x : M) :
e.toMatrix (Function.update v j x) = Matrix.updateColumn (e.toMatrix v) j (e.repr x) := by |
ext i' k
rw [Basis.toMatrix, Matrix.updateColumn_apply, e.toMatrix_apply]
split_ifs with h
· rw [h, update_same j x v]
· rw [update_noteq h]
|
/-
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 | 153 | 157 | theorem map₂_iSup_right (f : M →ₗ[R] N →ₗ[R] P) (s : Submodule R M) (t : ι → Submodule R N) :
map₂ f s (⨆ i, t i) = ⨆ i, map₂ f s (t i) := by |
suffices map₂ f (span R s) (⨆ i, span R (t i : Set N)) = ⨆ i, map₂ f (span R s) (span R (t i)) by
simpa only [span_eq] using this
simp_rw [map₂_span_span, ← span_iUnion, map₂_span_span, Set.image2_iUnion_right]
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 332 | 333 | theorem infsep_zero : s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞ := by |
rw [infsep, ENNReal.toReal_eq_zero_iff]
|
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 248 | 249 | theorem toList_get_zero (h : x ∈ p.support) :
(toList p x).get ⟨0, (length_toList_pos_of_mem_support _ _ h)⟩ = x := by | simp [toList]
|
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 443 | 446 | theorem angle_rev_eq_pi_div_two_of_oangle_eq_neg_pi_div_two {p₁ p₂ p₃ : P}
(h : ∡ p₁ p₂ p₃ = ↑(-π / 2)) : ∠ p₃ p₂ p₁ = π / 2 := by |
rw [angle_comm]
exact angle_eq_pi_div_two_of_oangle_eq_neg_pi_div_two h
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 798 | 802 | theorem exists_associated_prime_pow_of_unique_normalized_factor {p r : α}
(h : ∀ {m}, m ∈ normalizedFactors r → m = p) (hr : r ≠ 0) : ∃ i : ℕ, Associated (p ^ i) r := by |
use Multiset.card.toFun (normalizedFactors r)
have := UniqueFactorizationMonoid.normalizedFactors_prod hr
rwa [Multiset.eq_replicate_of_mem fun b => h, Multiset.prod_replicate] at this
|
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 481 | 483 | theorem restrict_sInf_eq_sInf_restrict (m : Set (OuterMeasure α)) {s : Set α} (hm : m.Nonempty) :
restrict s (sInf m) = sInf (restrict s '' m) := by |
simp only [sInf_eq_iInf, restrict_biInf, hm, iInf_image]
|
/-
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.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import m... | Mathlib/ModelTheory/Basic.lean | 1,007 | 1,011 | theorem comp_right_injective (h : M ≃[L] N) :
Function.Injective (fun f ↦ f.comp h : (N ≃[L] P) → (M ≃[L] P)) := by |
intro f g hfg
convert (congr_arg (fun r : (M ≃[L] P) ↦ r.comp h.symm) hfg) <;>
rw [comp_assoc, self_comp_symm, comp_refl]
|
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.c... | Mathlib/Analysis/Calculus/Deriv/Add.lean | 276 | 284 | theorem not_differentiableAt_abs_zero : ¬ DifferentiableAt ℝ (abs : ℝ → ℝ) 0 := by |
intro h
have h₁ : deriv abs (0 : ℝ) = 1 :=
(uniqueDiffOn_Ici _ _ Set.left_mem_Ici).eq_deriv _ h.hasDerivAt.hasDerivWithinAt <|
(hasDerivWithinAt_id _ _).congr_of_mem (fun _ h ↦ abs_of_nonneg h) Set.left_mem_Ici
have h₂ : deriv abs (0 : ℝ) = -1 :=
(uniqueDiffOn_Iic _ _ Set.right_mem_Iic).eq_deriv _ ... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 282 | 285 | theorem mul_adjugate_apply (A : Matrix n n α) (i j k) :
A i k * adjugate A k j = cramer Aᵀ (Pi.single k (A i k)) j := by |
erw [← smul_eq_mul, adjugate, of_apply, ← Pi.smul_apply, ← LinearMap.map_smul, ← Pi.single_smul',
smul_eq_mul, mul_one]
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Ker.lean | 112 | 113 | theorem ker_eq_bot' {f : F} : ker f = ⊥ ↔ ∀ m, f m = 0 → m = 0 := by |
simpa [disjoint_iff_inf_le] using disjoint_ker (f := f) (p := ⊤)
|
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 592 | 598 | theorem X_mul : X * p = p * X := by |
rcases p with ⟨⟩
-- Porting note: `ofFinsupp.injEq` is required.
simp only [X, ← ofFinsupp_single, ← ofFinsupp_mul, LinearMap.coe_mk, ofFinsupp.injEq]
-- Porting note: Was `ext`.
refine Finsupp.ext fun _ => ?_
simp [AddMonoidAlgebra.mul_apply, AddMonoidAlgebra.sum_single_index, add_comm]
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Data/Nat/Bitwise.lean | 395 | 396 | theorem xor_cancel_left (n m : ℕ) : n ^^^ (n ^^^ m) = m := by |
rw [← Nat.xor_assoc, xor_self, zero_xor]
|
/-
Copyright (c) 2022 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathli... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 561 | 563 | theorem mem_im_objs_iff (hφ : Function.Injective φ.obj) (d : D) :
d ∈ (im φ hφ).objs ↔ ∃ c : C, φ.obj c = d := by |
simp only [im, mem_map_objs_iff, mem_top_objs, true_and]
|
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