Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 202 | 205 | theorem inner_rightAngleRotationAux₁_right (x y : E) :
⟪x, o.rightAngleRotationAux₁ y⟫ = -ω x y := by |
rw [real_inner_comm]
simp [o.areaForm_swap y x]
|
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import probability.martingale.convergence from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Me... | Mathlib/Probability/Martingale/Convergence.lean | 156 | 183 | theorem Submartingale.upcrossings_ae_lt_top' [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
(hbdd : ∀ n, snorm (f n) 1 μ ≤ R) (hab : a < b) : ∀ᵐ ω ∂μ, upcrossings a b f ω < ∞ := by |
refine ae_lt_top (hf.adapted.measurable_upcrossings hab) ?_
have := hf.mul_lintegral_upcrossings_le_lintegral_pos_part a b
rw [mul_comm, ← ENNReal.le_div_iff_mul_le] at this
· refine (lt_of_le_of_lt this (ENNReal.div_lt_top ?_ ?_)).ne
· have hR' : ∀ n, ∫⁻ ω, ‖f n ω - a‖₊ ∂μ ≤ R + ‖a‖₊ * μ Set.univ := by
... |
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 455 | 459 | theorem limit.lift_post (c : Cone F) :
G.map (limit.lift F c) ≫ limit.post F G = limit.lift (F ⋙ G) (G.mapCone c) := by |
ext
rw [assoc, limit.post_π, ← G.map_comp, limit.lift_π, limit.lift_π]
rfl
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 82 | 84 | theorem continuous_cosh : Continuous cosh := by |
change Continuous fun z => (exp z + exp (-z)) / 2
continuity
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 951 | 955 | theorem MonoidHomClass.isometry_iff_norm [MonoidHomClass 𝓕 E F] (f : 𝓕) :
Isometry f ↔ ∀ x, ‖f x‖ = ‖x‖ := by |
simp only [isometry_iff_dist_eq, dist_eq_norm_div, ← map_div]
refine ⟨fun h x => ?_, fun h x y => h _⟩
simpa using h x 1
|
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 139 | 144 | theorem opow_le_opow_right {a b c : Ordinal} (h₁ : 0 < a) (h₂ : b ≤ c) : a ^ b ≤ a ^ c := by |
rcases lt_or_eq_of_le (one_le_iff_pos.2 h₁) with h₁ | h₁
· exact (opow_le_opow_iff_right h₁).2 h₂
· subst a
-- Porting note: `le_refl` is required.
simp only [one_opow, le_refl]
|
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 149 | 154 | theorem Sphere.secondInter_collinear (s : Sphere P) (p p' : P) :
Collinear ℝ ({p, p', s.secondInter p (p' -ᵥ p)} : Set P) := by |
rw [Set.pair_comm, Set.insert_comm]
exact
(collinear_insert_iff_of_mem_affineSpan (s.secondInter_vsub_mem_affineSpan _ _)).2
(collinear_pair ℝ _ _)
|
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 118 | 120 | theorem degree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
(cyclotomic' n R).degree = Nat.totient n := by |
simp only [degree_eq_natDegree (cyclotomic'_ne_zero n R), natDegree_cyclotomic' h]
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace List
variable {α β : Type*}
section FormPerm
variable [DecidableEq α] (l :... | Mathlib/GroupTheory/Perm/List.lean | 396 | 405 | theorem formPerm_eq_one_iff (hl : Nodup l) : formPerm l = 1 ↔ l.length ≤ 1 := by |
cases' l with hd tl
· simp
· rw [← formPerm_apply_mem_eq_self_iff _ hl hd (mem_cons_self _ _)]
constructor
· simp (config := { contextual := true })
· intro h
simp only [(hd :: tl).formPerm_apply_mem_eq_self_iff hl hd (mem_cons_self hd tl),
add_le_iff_nonpos_left, length, nonpos_iff_eq_... |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 101 | 104 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by |
classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩
|
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 88 | 88 | theorem absNorm_one : absNorm (1 : FractionalIdeal R⁰ K) = 1 := by | convert absNorm.map_one'
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Multinomial
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped C... | Mathlib/Analysis/Calculus/ContDiff/Bounds.lean | 224 | 231 | theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_of_le_one
(B : E →L[𝕜] F →L[𝕜] G) {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D}
(hf : ContDiffOn 𝕜 N f s) (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ}
(hn : (n : ℕ∞) ≤ N) (hB : ‖B‖ ≤ 1) : ‖iteratedFDer... |
apply (B.norm_iteratedFDerivWithin_le_of_bilinear hf hg hs hx hn).trans
exact mul_le_of_le_one_left (by positivity) hB
|
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 730 | 731 | theorem dropn_think (s : WSeq α) (n) : drop (think s) n = (drop s n).think := by |
induction n <;> simp [*, drop]
|
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.IsLocalHomeomorph
#align_import topology.instances.add_circle from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/AddCircle.lean | 175 | 176 | theorem coe_add_period (x : 𝕜) : ((x + p : 𝕜) : AddCircle p) = x := by |
rw [coe_add, ← eq_sub_iff_add_eq', sub_self, coe_period]
|
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
assert_not_exists MonoidWithZero
open Prod Decidable Function
namespace Nat
-- Porting note: no pp_nodot
--@[pp_nodot]
def pair (a b : ... | Mathlib/Data/Nat/Pairing.lean | 114 | 114 | theorem left_le_pair (a b : ℕ) : a ≤ pair a b := by | simpa using unpair_left_le (pair a b)
|
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.manifold.cont_mdiff from "leanprover-community/mathlib"@"e5ab837fc252451f3eb9124ae6e7b6f57455e7b9"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scope... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 466 | 470 | theorem contMDiffAt_iff_target_of_mem_source {x : M} {y : M'} (hy : f x ∈ (chartAt H' y).source) :
ContMDiffAt I I' n f x ↔
ContinuousAt f x ∧ ContMDiffAt I 𝓘(𝕜, E') n (extChartAt I' y ∘ f) x := by |
rw [ContMDiffAt, contMDiffWithinAt_iff_target_of_mem_source hy, continuousWithinAt_univ,
ContMDiffAt]
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 106 | 110 | theorem snorm_condexpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) :
snorm (F := E) (condexpL2 E 𝕜 hm f) 2 μ ≤ snorm f 2 μ := by |
rw [lpMeas_coe, ← ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), ←
Lp.norm_def, ← Lp.norm_def, Submodule.norm_coe]
exact norm_condexpL2_le hm f
|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 178 | 179 | theorem frontier_Iio' {a : α} (ha : (Iio a).Nonempty) : frontier (Iio a) = {a} := by |
simp [frontier, closure_Iio' ha, Iic_diff_Iio, Icc_self]
|
import Batteries.Classes.Order
import Batteries.Control.ForInStep.Basic
namespace Batteries
namespace BinomialHeap
namespace Imp
inductive HeapNode (α : Type u) where
| nil : HeapNode α
| node (a : α) (child sibling : HeapNode α) : HeapNode α
deriving Repr
@[simp] def HeapNode.realSize : HeapNode α → ... | .lake/packages/batteries/Batteries/Data/BinomialHeap/Basic.lean | 205 | 221 | theorem Heap.realSize_merge (le) (s₁ s₂ : Heap α) :
(s₁.merge le s₂).realSize = s₁.realSize + s₂.realSize := by |
unfold merge; split
· simp
· simp
· next r₁ a₁ n₁ t₁ r₂ a₂ n₂ t₂ =>
have IH₁ r a n := realSize_merge le t₁ (cons r a n t₂)
have IH₂ r a n := realSize_merge le (cons r a n t₁) t₂
have IH₃ := realSize_merge le t₁ t₂
split; · simp [IH₁, Nat.add_assoc]
split; · simp [IH₂, Nat.add_assoc, Nat.add... |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 739 | 740 | theorem cons_map_snd_darts {u v : V} (p : G.Walk u v) : (u :: p.darts.map (·.snd)) = p.support := by |
induction p <;> simp! [*]
|
import Mathlib.Analysis.SpecialFunctions.Log.Base
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.doubling from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filter Metric MeasureTheory TopologicalSpace ENNReal NN... | Mathlib/MeasureTheory/Measure/Doubling.lean | 139 | 143 | theorem eventually_measure_le_scaling_constant_mul' (K : ℝ) (hK : 0 < K) :
∀ᶠ r in 𝓝[>] 0, ∀ x,
μ (closedBall x r) ≤ scalingConstantOf μ K⁻¹ * μ (closedBall x (K * r)) := by |
convert eventually_nhdsWithin_pos_mul_left hK (eventually_measure_le_scaling_constant_mul μ K⁻¹)
simp [inv_mul_cancel_left₀ hK.ne']
|
import Mathlib.Algebra.Group.Support
import Mathlib.Order.WellFoundedSet
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
open scoped Classical
noncomputable section
@[ext]
structure ... | Mathlib/RingTheory/HahnSeries/Basic.lean | 357 | 367 | theorem embDomain_coeff {f : Γ ↪o Γ'} {x : HahnSeries Γ R} {a : Γ} :
(embDomain f x).coeff (f a) = x.coeff a := by |
rw [embDomain]
dsimp only
by_cases ha : a ∈ x.support
· rw [dif_pos (Set.mem_image_of_mem f ha)]
exact congr rfl (f.injective (Classical.choose_spec (Set.mem_image_of_mem f ha)).2)
· rw [dif_neg, Classical.not_not.1 fun c => ha ((mem_support _ _).2 c)]
contrapose! ha
obtain ⟨b, hb1, hb2⟩ := (Set.... |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open sc... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 535 | 554 | theorem exists_upperSemicontinuous_lt_integral_gt [SigmaFinite μ] (f : α → ℝ) (hf : Integrable f μ)
{ε : ℝ} (εpos : 0 < ε) :
∃ g : α → EReal,
(∀ x, (g x : EReal) < f x) ∧
UpperSemicontinuous g ∧
Integrable (fun x => EReal.toReal (g x)) μ ∧
(∀ᵐ x ∂μ, ⊥ < g x) ∧ (∫ x, f x ∂μ) < (∫ x, EReal... |
rcases exists_lt_lowerSemicontinuous_integral_lt (fun x => -f x) hf.neg εpos with
⟨g, g_lt_f, gcont, g_integrable, g_lt_top, gint⟩
refine ⟨fun x => -g x, ?_, ?_, ?_, ?_, ?_⟩
· exact fun x => EReal.neg_lt_iff_neg_lt.1 (by simpa only [EReal.coe_neg] using g_lt_f x)
· exact
continuous_neg.comp_lowerSemi... |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.l2Space
import Mathlib.MeasureTheory.Function.ContinuousMapDense
import Mathlib.MeasureTheory.Function.L2Space
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.M... | Mathlib/Analysis/Fourier/AddCircle.lean | 247 | 250 | theorem span_fourier_closure_eq_top : (span ℂ (range <| @fourier T)).topologicalClosure = ⊤ := by |
rw [← fourierSubalgebra_coe]
exact congr_arg (Subalgebra.toSubmodule <| StarSubalgebra.toSubalgebra ·)
fourierSubalgebra_closure_eq_top
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 468 | 471 | theorem norm_div_sin_oangle_sub_left_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖y‖ / Real.Angle.sin (o.oangle (x - y) x) = ‖x - y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
exact (-o).norm_div_sin_oangle_sub_right_of_oangle_eq_pi_div_two h
|
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : ℕ)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 79 | 80 | theorem map_valEmbedding_Icc : (Icc a b).map Fin.valEmbedding = Icc ↑a ↑b := by |
simp [Icc_eq_finset_subtype, Finset.fin, Finset.map_map, Icc_filter_lt_of_lt_right]
|
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Topology.Algebra.InfiniteSum.NatInt
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Order.MonotoneConvergence
#align_import topology.algebra.infinite_sum.order from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
... | Mathlib/Topology/Algebra/InfiniteSum/Order.lean | 289 | 298 | theorem multipliable_mabs_iff [LinearOrderedCommGroup α] [UniformSpace α] [UniformGroup α]
[CompleteSpace α] {f : ι → α} : (Multipliable fun x ↦ mabs (f x)) ↔ Multipliable f :=
let s := { x | 1 ≤ f x }
have h1 : ∀ x : s, mabs (f x) = f x := fun x ↦ mabs_of_one_le x.2
have h2 : ∀ x : ↑sᶜ, mabs (f x) = (f x)⁻¹ ... | simp only [h1, h2]
_ ↔ Multipliable f := by simp only [multipliable_inv_iff, multipliable_subtype_and_compl]
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
open MeasureTheory
section DominatedConvergenceTheorem
open Set Filter TopologicalSpace ENNReal
open scoped Topology
namespace MeasureTheory
variable {α E G: Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 79 | 104 | theorem hasSum_integral_of_dominated_convergence {ι} [Countable ι] {F : ι → α → G} {f : α → G}
(bound : ι → α → ℝ) (hF_meas : ∀ n, AEStronglyMeasurable (F n) μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound n a)
(bound_summable : ∀ᵐ a ∂μ, Summable fun n => bound n a)
(bound_integrable : Integrable (fun a => ... |
have hb_nonneg : ∀ᵐ a ∂μ, ∀ n, 0 ≤ bound n a :=
eventually_countable_forall.2 fun n => (h_bound n).mono fun a => (norm_nonneg _).trans
have hb_le_tsum : ∀ n, bound n ≤ᵐ[μ] fun a => ∑' n, bound n a := by
intro n
filter_upwards [hb_nonneg, bound_summable]
with _ ha0 ha_sum using le_tsum ha_sum _ fu... |
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 121 | 128 | theorem of_equiv [FormallyUnramified R A] (e : A ≃ₐ[R] B) :
FormallyUnramified R B := by |
constructor
intro C _ _ I hI f₁ f₂ e'
rw [← f₁.comp_id, ← f₂.comp_id, ← e.comp_symm, ← AlgHom.comp_assoc, ← AlgHom.comp_assoc]
congr 1
refine FormallyUnramified.comp_injective I hI ?_
rw [← AlgHom.comp_assoc, e', AlgHom.comp_assoc]
|
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 332 | 338 | theorem fst_embedding_of_right_embedding {X Y S : TopCat} (f : X ⟶ S) {g : Y ⟶ S}
(H : Embedding g) : Embedding <| ⇑(pullback.fst : pullback f g ⟶ X) := by |
convert (homeoOfIso (asIso (pullback.fst : pullback f (𝟙 S) ⟶ _))).embedding.comp
(pullback_map_embedding_of_embeddings (i₁ := 𝟙 X)
f g f (𝟙 _) (homeoOfIso (Iso.refl _)).embedding H (𝟙 _) rfl (by simp))
erw [← coe_comp]
simp
|
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.Subring.Basic
import Mathlib.Topology.Algebra.Group.Basic
#align_import topology.algebra.ring.basic from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpac... | Mathlib/Topology/Algebra/Ring/Basic.lean | 63 | 66 | theorem TopologicalSemiring.continuousNeg_of_mul [TopologicalSpace α] [NonAssocRing α]
[ContinuousMul α] : ContinuousNeg α where
continuous_neg := by |
simpa using (continuous_const.mul continuous_id : Continuous fun x : α => -1 * x)
|
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 65 | 68 | theorem ofReal_add {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) :
ENNReal.ofReal (p + q) = ENNReal.ofReal p + ENNReal.ofReal q := by |
rw [ENNReal.ofReal, ENNReal.ofReal, ENNReal.ofReal, ← coe_add, coe_inj,
Real.toNNReal_add hp hq]
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 195 | 214 | theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
(hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
(h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
(hk : (k : ℕ∞) ≤ N) :
(iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k ... |
induction' k with k IH
· ext1 x
simp_rw [iteratedFDeriv_zero_eq_comp]
exact (continuousMultilinearCurryFin0 𝕜 E F).symm.toContinuousLinearEquiv.map_tsum
· have h'k : (k : ℕ∞) < N := lt_of_lt_of_le (WithTop.coe_lt_coe.2 (Nat.lt_succ_self _)) hk
have A : Summable fun n => iteratedFDeriv 𝕜 k (f n) 0 :... |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 146 | 148 | theorem inv_pos_lt_iff_one_lt_mul' (ha : 0 < a) : a⁻¹ < b ↔ 1 < a * b := by |
rw [inv_eq_one_div]
exact div_lt_iff' ha
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 92 | 95 | theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by |
ext v
refine ext_inner_left 𝕜 fun w => ?_
rw [adjointAux_inner_right, adjointAux_inner_left]
|
import Mathlib.Algebra.Module.Torsion
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
... | Mathlib/LinearAlgebra/Dimension/Finite.lean | 82 | 84 | theorem rank_pos_iff_exists_ne_zero : 0 < Module.rank R M ↔ ∃ x : M, x ≠ 0 := by |
rw [← not_iff_not]
simpa using rank_zero_iff_forall_zero
|
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 113 | 114 | theorem image_const_add_Ici : (fun x => a + x) '' Ici b = Ici (a + b) := by |
simp only [add_comm a, image_add_const_Ici]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 371 | 382 | theorem UniformContinuousOn.tendstoUniformlyOn [UniformSpace α] [UniformSpace γ] {x : α} {U : Set α}
{V : Set β} {F : α → β → γ} (hF : UniformContinuousOn (↿F) (U ×ˢ V)) (hU : x ∈ U) :
TendstoUniformlyOn F (F x) (𝓝[U] x) V := by |
set φ := fun q : α × β => ((x, q.2), q)
rw [tendstoUniformlyOn_iff_tendsto]
change Tendsto (Prod.map (↿F) ↿F ∘ φ) (𝓝[U] x ×ˢ 𝓟 V) (𝓤 γ)
simp only [nhdsWithin, SProd.sprod, Filter.prod, comap_inf, inf_assoc, comap_principal,
inf_principal]
refine hF.comp (Tendsto.inf ?_ <| tendsto_principal_principal.2... |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.CategoryTheory.Elementwise
import Ma... | Mathlib/CategoryTheory/Limits/Shapes/Types.lean | 543 | 544 | theorem equalizerIso_hom_comp_subtype : (equalizerIso g h).hom ≫ Subtype.val = equalizer.ι g h := by |
rfl
|
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.liouville_with from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Filter Metric Real Set
open sc... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 287 | 287 | theorem rat_sub_iff : LiouvilleWith p (r - x) ↔ LiouvilleWith p x := by | simp [sub_eq_add_neg]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 58 | 62 | theorem ne_zero_of_ne_zero_of_monic (hp : p ≠ 0) (hq : Monic q) : q ≠ 0 := by |
rintro rfl
rw [Monic.def, leadingCoeff_zero] at hq
rw [← mul_one p, ← C_1, ← hq, C_0, mul_zero] at hp
exact hp rfl
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Data.List.Sort
import Mathlib.Data.List.Chain
#align_import data.nat.factors from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Bool Subtype
open Nat
namespac... | Mathlib/Data/Nat/Factors.lean | 49 | 49 | theorem factors_one : factors 1 = [] := by | rw [factors]
|
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Integral.DominatedConve... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,323 | 1,332 | theorem intervalIntegrable_deriv_of_nonneg (hcont : ContinuousOn g (uIcc a b))
(hderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x)
(hpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x) : IntervalIntegrable g' volume a b := by |
rcases le_total a b with hab | hab
· simp only [uIcc_of_le, min_eq_left, max_eq_right, hab, IntervalIntegrable, hab,
Ioc_eq_empty_of_le, integrableOn_empty, and_true_iff] at hcont hderiv hpos ⊢
exact integrableOn_deriv_of_nonneg hcont hderiv hpos
· simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab,... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 90 | 92 | theorem prod_univ_castSucc [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) :
∏ i, f i = (∏ i : Fin n, f (Fin.castSucc i)) * f (last n) := by |
simpa [mul_comm] using prod_univ_succAbove f (last n)
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 82 | 83 | theorem vars_monomial (h : r ≠ 0) : (monomial s r).vars = s.support := by |
classical rw [vars_def, degrees_monomial_eq _ _ h, Finsupp.toFinset_toMultiset]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 395 | 401 | theorem exists_coprime_of_gcd_pos {a : ℤ√d} (hgcd : 0 < Int.gcd a.re a.im) :
∃ b : ℤ√d, a = ((Int.gcd a.re a.im : ℤ) : ℤ√d) * b ∧ IsCoprime b.re b.im := by |
obtain ⟨re, im, H1, Hre, Him⟩ := Int.exists_gcd_one hgcd
rw [mul_comm] at Hre Him
refine ⟨⟨re, im⟩, ?_, ?_⟩
· rw [smul_val, ← Hre, ← Him]
· rw [← Int.gcd_eq_one_iff_coprime, H1]
|
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 126 | 132 | theorem pumping_lemma [Fintype σ] {x : List α} (hx : x ∈ M.accepts)
(hlen : Fintype.card (Set σ) ≤ List.length x) :
∃ a b c,
x = a ++ b ++ c ∧
a.length + b.length ≤ Fintype.card (Set σ) ∧ b ≠ [] ∧ {a} * {b}∗ * {c} ≤ M.accepts := by |
rw [← toDFA_correct] at hx ⊢
exact M.toDFA.pumping_lemma hx hlen
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 346 | 349 | theorem sin_eq_real_sin_iff_eq_or_add_eq_pi {θ : Angle} {ψ : ℝ} :
sin θ = Real.sin ψ ↔ θ = ψ ∨ θ + ψ = π := by |
induction θ using Real.Angle.induction_on
exact sin_eq_iff_coe_eq_or_add_eq_pi
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 185 | 188 | theorem HasStrictDerivAt.comp_hasStrictFDerivAt_of_eq {f : E → 𝕜'} {f' : E →L[𝕜] 𝕜'} (x)
(hh : HasStrictDerivAt h₂ h₂' y) (hf : HasStrictFDerivAt f f' x) (hy : y = f x) :
HasStrictFDerivAt (h₂ ∘ f) (h₂' • f') x := by |
rw [hy] at hh; exact hh.comp_hasStrictFDerivAt x hf
|
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 110 | 112 | theorem measure_sUnion_null_iff {S : Set (Set α)} (hS : S.Countable) :
μ (⋃₀ S) = 0 ↔ ∀ s ∈ S, μ s = 0 := by |
rw [sUnion_eq_biUnion, measure_biUnion_null_iff hS]
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,635 | 1,648 | theorem sup_eq_lsub_iff_succ {ι : Type u} (f : ι → Ordinal.{max u v}) :
sup.{_, v} f = lsub.{_, v} f ↔ ∀ a < lsub.{_, v} f, succ a < lsub.{_, v} f := by |
refine ⟨fun h => ?_, fun hf => le_antisymm (sup_le_lsub f) (lsub_le fun i => ?_)⟩
· rw [← h]
exact fun a => sup_not_succ_of_ne_sup fun i => (lsub_le_iff.1 (le_of_eq h.symm) i).ne
by_contra! hle
have heq := (sup_succ_eq_lsub f).2 ⟨i, le_antisymm (le_sup _ _) hle⟩
have :=
hf _
(by
rw [← h... |
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 381 | 387 | theorem T0Space.of_cover (h : ∀ x y, Inseparable x y → ∃ s : Set X, x ∈ s ∧ y ∈ s ∧ T0Space s) :
T0Space X := by |
refine ⟨fun x y hxy => ?_⟩
rcases h x y hxy with ⟨s, hxs, hys, hs⟩
lift x to s using hxs; lift y to s using hys
rw [← subtype_inseparable_iff] at hxy
exact congr_arg Subtype.val hxy.eq
|
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 172 | 177 | theorem next_cons_concat (y : α) (hy : x ≠ y) (hx : x ∉ l)
(h : x ∈ y :: l ++ [x] := mem_append_right _ (mem_singleton_self x)) :
next (y :: l ++ [x]) x h = y := by |
rw [next, nextOr_concat]
· rfl
· simp [hy, hx]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 142 | 148 | theorem smul_induction_on {p : M → Prop} {x} (H : x ∈ I • N) (smul : ∀ r ∈ I, ∀ n ∈ N, p (r • n))
(add : ∀ x y, p x → p y → p (x + y)) : p x := by |
have H0 : p 0 := by simpa only [zero_smul] using smul 0 I.zero_mem 0 N.zero_mem
refine Submodule.iSup_induction (x := x) _ H ?_ H0 add
rintro ⟨i, hi⟩ m ⟨j, hj, hj'⟩
rw [← hj']
exact smul _ hi _ hj
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 155 | 158 | theorem deriv.scomp_of_eq
(hg : DifferentiableAt 𝕜' g₁ y) (hh : DifferentiableAt 𝕜 h x) (hy : y = h x) :
deriv (g₁ ∘ h) x = deriv h x • deriv g₁ (h x) := by |
rw [hy] at hg; exact deriv.scomp x hg hh
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 501 | 503 | theorem eq_coe_norm_of_nonneg {z : ℂ} (hz : 0 ≤ z) : z = ↑‖z‖ := by |
lift z to ℝ using hz.2.symm
rw [norm_eq_abs, abs_ofReal, _root_.abs_of_nonneg (id hz.1 : 0 ≤ z)]
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
assert_not_exists HasFDerivAt
assert_not_exists ConformalAt
noncom... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 134 | 137 | theorem angle_self {x : V} (hx : x ≠ 0) : angle x x = 0 := by |
unfold angle
rw [← real_inner_self_eq_norm_mul_norm, div_self (inner_self_ne_zero.2 hx : ⟪x, x⟫ ≠ 0),
Real.arccos_one]
|
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
namespace Metric
#align metric.bounded Bornology.I... | Mathlib/Topology/MetricSpace/Bounded.lean | 209 | 215 | theorem isBounded_range_of_tendsto_cofinite_uniformity {f : β → α}
(hf : Tendsto (Prod.map f f) (.cofinite ×ˢ .cofinite) (𝓤 α)) : IsBounded (range f) := by |
rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
⟨s, hsf, hs1⟩
rw [← image_union_image_compl_eq_range]
refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩)
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 289 | 295 | theorem integrable_indicatorConstLp {E} [NormedAddCommGroup E] {p : ℝ≥0∞} {s : Set α}
(hs : MeasurableSet s) (hμs : μ s ≠ ∞) (c : E) :
Integrable (indicatorConstLp p hs hμs c) μ := by |
rw [integrable_congr indicatorConstLp_coeFn, integrable_indicator_iff hs, IntegrableOn,
integrable_const_iff, lt_top_iff_ne_top]
right
simpa only [Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply] using hμs
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 126 | 132 | theorem projectiveSeminorm_tprod_le (m : Π i, E i) :
projectiveSeminorm (⨂ₜ[𝕜] i, m i) ≤ ∏ i, ‖m i‖ := by |
rw [projectiveSeminorm_apply]
convert ciInf_le (bddBelow_projectiveSemiNormAux _) ⟨[((1 : 𝕜), m)] ,?_⟩
· simp only [projectiveSeminormAux, Function.comp_apply, List.map_cons, norm_one, one_mul,
List.map_nil, List.sum_cons, List.sum_nil, add_zero]
· rw [mem_lifts_iff, List.map_singleton, List.sum_singleton... |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b := by |
simp only [lineMap_apply_module]
rw [← le_sub_iff_add_le, add_sub_assoc, ← sub_le_iff_le_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]
|
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 159 | 162 | theorem Integrable.ae_eq_of_withDensityᵥ_eq {f g : α → E} (hf : Integrable f μ)
(hg : Integrable g μ) (hfg : μ.withDensityᵥ f = μ.withDensityᵥ g) : f =ᵐ[μ] g := by |
refine hf.ae_eq_of_forall_setIntegral_eq f g hg fun i hi _ => ?_
rw [← withDensityᵥ_apply hf hi, hfg, withDensityᵥ_apply hg hi]
|
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
| Mathlib/Algebra/Group/Ext.lean | 38 | 51 | theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Monoid M⦄
(h_mul : (letI := m₁; HMul.hMul : M → M → M) = (letI := m₂; HMul.hMul : M → M → M)) :
m₁ = m₂ := by |
have : m₁.toMulOneClass = m₂.toMulOneClass := MulOneClass.ext h_mul
have h₁ : m₁.one = m₂.one := congr_arg (·.one) this
let f : @MonoidHom M M m₁.toMulOneClass m₂.toMulOneClass :=
@MonoidHom.mk _ _ (_) _ (@OneHom.mk _ _ (_) _ id h₁)
(fun x y => congr_fun (congr_fun h_mul x) y)
have : m₁.npow = m₂.npo... |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 250 | 261 | theorem prev_mem (h : x ∈ l) : l.prev x h ∈ l := by |
cases' l with hd tl
· simp at h
induction' tl with hd' tl hl generalizing hd
· simp
· by_cases hx : x = hd
· simp only [hx, prev_cons_cons_eq]
exact mem_cons_of_mem _ (getLast_mem _)
· rw [prev, dif_neg hx]
split_ifs with hm
· exact mem_cons_self _ _
· exact mem_cons_of_mem _ ... |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.between from "leanprover-community/mathlib"@"571e13cacbed7bf042fd3058c... | Mathlib/Analysis/Convex/Between.lean | 354 | 374 | theorem wbtw_swap_left_iff [NoZeroSMulDivisors R V] {x y : P} (z : P) :
Wbtw R x y z ∧ Wbtw R y x z ↔ x = y := by |
constructor
· rintro ⟨hxyz, hyxz⟩
rcases hxyz with ⟨ty, hty, rfl⟩
rcases hyxz with ⟨tx, htx, hx⟩
rw [lineMap_apply, lineMap_apply, ← add_vadd] at hx
rw [← @vsub_eq_zero_iff_eq V, vadd_vsub, vsub_vadd_eq_vsub_sub, smul_sub, smul_smul, ← sub_smul,
← add_smul, smul_eq_zero] at hx
rcases hx w... |
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 1,907 | 1,909 | theorem diff_singleton_subset_iff {x : α} {s t : Set α} : s \ {x} ⊆ t ↔ s ⊆ insert x t := by |
rw [← union_singleton, union_comm]
apply diff_subset_iff
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 817 | 818 | theorem inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
⟪y, x⟫ = 0 := by | rw [real_inner_comm, o.inner_eq_zero_of_oangle_eq_neg_pi_div_two h]
|
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 518 | 520 | theorem coe_equiv_snd_eq_one_iff_mem {g : G} (h1 : 1 ∈ T) :
((hST.equiv g).snd : G) = 1 ↔ g ∈ S := by |
rw [equiv_snd_eq_inv_mul, inv_mul_eq_one, equiv_fst_eq_self_iff_mem _ h1]
|
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 360 | 361 | theorem IsLindelof.union (hs : IsLindelof s) (ht : IsLindelof t) : IsLindelof (s ∪ t) := by |
rw [union_eq_iUnion]; exact isLindelof_iUnion fun b => by cases b <;> assumption
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 176 | 178 | theorem toLeftMovesNim_one_symm (i) :
(@toLeftMovesNim 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by |
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 442 | 456 | theorem getVert_reverse {u v : V} (p : G.Walk u v) (i : ℕ) :
p.reverse.getVert i = p.getVert (p.length - i) := by |
induction p with
| nil => rfl
| cons h p ih =>
simp only [reverse_cons, getVert_append, length_reverse, ih, length_cons]
split_ifs
next hi =>
rw [Nat.succ_sub hi.le]
simp [getVert]
next hi =>
obtain rfl | hi' := Nat.eq_or_lt_of_not_lt hi
· simp [getVert]
· rw [Nat.eq... |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 167 | 170 | theorem mem_permutationsAux2' {t : α} {ts : List α} {ys : List α} {l : List α} :
l ∈ (permutationsAux2 t ts [] ys id).2 ↔
∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l = l₁ ++ t :: l₂ ++ ts := by |
rw [show @id (List α) = ([] ++ ·) by funext _; rfl]; apply mem_permutationsAux2
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 654 | 654 | theorem forall_mem_range {p : α → Prop} : (∀ a ∈ range f, p a) ↔ ∀ i, p (f i) := by | simp
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
section S... | Mathlib/MeasureTheory/Function/LpSeminorm/CompareExp.lean | 88 | 92 | theorem snorm'_le_snorm'_of_exponent_le {p q : ℝ} (hp0_lt : 0 < p)
(hpq : p ≤ q) (μ : Measure α) [IsProbabilityMeasure μ] (hf : AEStronglyMeasurable f μ) :
snorm' f p μ ≤ snorm' f q μ := by |
have h_le_μ := snorm'_le_snorm'_mul_rpow_measure_univ hp0_lt hpq hf
rwa [measure_univ, ENNReal.one_rpow, mul_one] at h_le_μ
|
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 93 | 95 | theorem invFun_add (M N : Matrix n n A) :
invFun R A n (M + N) = invFun R A n M + invFun R A n N := by |
simp [invFun, add_tmul, Finset.sum_add_distrib]
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 116 | 118 | theorem to_leftMoves_one_toPGame_symm (i) :
(@toLeftMovesToPGame 1).symm i = ⟨0, Set.mem_Iio.mpr zero_lt_one⟩ := by |
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 72 | 73 | theorem natDegree_le_iff_coeff_eq_zero : p.natDegree ≤ n ↔ ∀ N : ℕ, n < N → p.coeff N = 0 := by |
simp_rw [natDegree_le_iff_degree_le, degree_le_iff_coeff_zero, Nat.cast_lt]
|
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 354 | 367 | theorem Real.Gamma_one_half_eq : Real.Gamma (1 / 2) = √π := by |
rw [Gamma_eq_integral one_half_pos, ← integral_comp_rpow_Ioi_of_pos zero_lt_two]
convert congr_arg (fun x : ℝ => 2 * x) (integral_gaussian_Ioi 1) using 1
· rw [← integral_mul_left]
refine setIntegral_congr measurableSet_Ioi fun x hx => ?_
dsimp only
have : (x ^ (2 : ℝ)) ^ (1 / (2 : ℝ) - 1) = x⁻¹ := b... |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 549 | 549 | theorem mem_ball' : y ∈ ball x ε ↔ edist x y < ε := by | rw [edist_comm, mem_ball]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "leanprover-community/mathlib"@"80c43012d26f63026d362c3aba28f3c3bafb07e6"
open Function
structure Part.{u} (α : Type u) : Type u where
Dom : Prop
get : Dom → α
#align part... | Mathlib/Data/Part.lean | 405 | 415 | theorem le_total_of_le_of_le {x y : Part α} (z : Part α) (hx : x ≤ z) (hy : y ≤ z) :
x ≤ y ∨ y ≤ x := by |
rcases Part.eq_none_or_eq_some x with (h | ⟨b, h₀⟩)
· rw [h]
left
apply OrderBot.bot_le _
right; intro b' h₁
rw [Part.eq_some_iff] at h₀
have hx := hx _ h₀; have hy := hy _ h₁
have hx := Part.mem_unique hx hy; subst hx
exact h₀
|
import Mathlib.Algebra.Group.Commutator
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Bracket
import Mathlib.GroupTheory.Subgroup.Centralizer
import Mathlib.Tactic.Group
#align_import group_theory.commutator from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
variable... | Mathlib/GroupTheory/Commutator.lean | 61 | 62 | theorem commutatorElement_inv : ⁅g₁, g₂⁆⁻¹ = ⁅g₂, g₁⁆ := by |
simp_rw [commutatorElement_def, mul_inv_rev, inv_inv, mul_assoc]
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 517 | 524 | theorem image_inter_preimage (f : α → β) (s : Set α) (t : Set β) :
f '' (s ∩ f ⁻¹' t) = f '' s ∩ t := by |
apply Subset.antisymm
· calc
f '' (s ∩ f ⁻¹' t) ⊆ f '' s ∩ f '' (f ⁻¹' t) := image_inter_subset _ _ _
_ ⊆ f '' s ∩ t := inter_subset_inter_right _ (image_preimage_subset f t)
· rintro _ ⟨⟨x, h', rfl⟩, h⟩
exact ⟨x, ⟨h', h⟩, rfl⟩
|
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 132 | 140 | theorem image_gaugeRescaleHomeomorph_closure {s t : Set E}
(hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0) (hsb : IsVonNBounded ℝ s)
(htc : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
gaugeRescaleHomeomorph s t hsc hs₀ hsb htc ht₀ htb '' closure s = closure t := by |
refine Subset.antisymm (mapsTo_gaugeRescale_closure hsc hs₀ htc
(mem_of_mem_nhds ht₀) (absorbent_nhds_zero ht₀)).image_subset ?_
rw [← Homeomorph.preimage_symm, ← image_subset_iff]
exact (mapsTo_gaugeRescale_closure htc ht₀ hsc
(mem_of_mem_nhds hs₀) (absorbent_nhds_zero hs₀)).image_subset
|
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Finite.Set
#align_import combinatorics.simple_graph.ends.defs from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
universe u
variable {V : Type u} (G : SimpleGraph V... | Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean | 178 | 180 | theorem subset_hom (C : G.ComponentCompl L) (h : K ⊆ L) : (C : Set V) ⊆ (C.hom h : Set V) := by |
rintro c ⟨cL, rfl⟩
exact ⟨fun h' => cL (h h'), rfl⟩
|
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {α : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace α} (μ : @Measure α m0) (hm : m ≤ m0) : @Measure α m :=
@OuterMeasure.toMeasure α m μ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 49 | 50 | theorem zero_trim (hm : m ≤ m0) : (0 : Measure α).trim hm = (0 : @Measure α m) := by |
simp [Measure.trim, @OuterMeasure.toMeasure_zero _ m]
|
import Mathlib.Algebra.Group.Action.Defs
#align_import group_theory.group_action.sum from "leanprover-community/mathlib"@"f1a2caaf51ef593799107fe9a8d5e411599f3996"
variable {M N P α β γ : Type*}
namespace Sum
section SMul
variable [SMul M α] [SMul M β] [SMul N α] [SMul N β] (a : M) (b : α) (c : β)
(x : Sum α... | Mathlib/GroupTheory/GroupAction/Sum.lean | 56 | 56 | theorem smul_swap : (a • x).swap = a • x.swap := by | cases x <;> rfl
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 2,256 | 2,258 | theorem iSup_iUnion (s : ι → Set α) (f : α → β) : ⨆ a ∈ ⋃ i, s i, f a = ⨆ (i) (a ∈ s i), f a := by |
rw [iSup_comm]
simp_rw [mem_iUnion, iSup_exists]
|
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 797 | 823 | theorem subset_countable_closure_of_almost_dense_set (s : Set α)
(hs : ∀ ε > 0, ∃ t : Set α, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε) :
∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by |
rcases s.eq_empty_or_nonempty with (rfl | ⟨x₀, hx₀⟩)
· exact ⟨∅, empty_subset _, countable_empty, empty_subset _⟩
choose! T hTc hsT using fun n : ℕ => hs n⁻¹ (by simp)
have : ∀ r x, ∃ y ∈ s, closedBall x r ∩ s ⊆ closedBall y (r * 2) := fun r x => by
rcases (closedBall x r ∩ s).eq_empty_or_nonempty with (he... |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 150 | 153 | theorem coeff_monomial_ne {m n : σ →₀ ℕ} (h : m ≠ n) (a : R) : coeff R m (monomial R n a) = 0 := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_ne R (fun _ ↦ R) _ _ h a
|
import Mathlib.CategoryTheory.Action
import Mathlib.Combinatorics.Quiver.Arborescence
import Mathlib.Combinatorics.Quiver.ConnectedComponent
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
#align_import group_theory.nielsen_schreier from "leanprover-community/mathlib"@"1bda4fc53de6ade5ab9da36f2192e24e2084a2ce"
n... | Mathlib/GroupTheory/FreeGroup/NielsenSchreier.lean | 178 | 179 | theorem treeHom_eq {a : G} (p : Path (root T) a) : treeHom T a = homOfPath T p := by |
rw [treeHom, Unique.default_eq]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 108 | 111 | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by |
let ⟨a, b, H⟩ := H1
rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
|
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 94 | 102 | theorem thickenedIndicatorAux_zero {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) {x : α}
(x_out : x ∉ thickening δ E) : thickenedIndicatorAux δ E x = 0 := by |
rw [thickening, mem_setOf_eq, not_lt] at x_out
unfold thickenedIndicatorAux
apply le_antisymm _ bot_le
have key := tsub_le_tsub
(@rfl _ (1 : ℝ≥0∞)).le (ENNReal.div_le_div x_out (@rfl _ (ENNReal.ofReal δ : ℝ≥0∞)).le)
rw [ENNReal.div_self (ne_of_gt (ENNReal.ofReal_pos.mpr δ_pos)) ofReal_ne_top] at key
si... |
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 189 | 192 | theorem sheafify_hom_ext {P Q : Cᵒᵖ ⥤ D} (η γ : sheafify J P ⟶ Q) (hQ : Presheaf.IsSheaf J Q)
(h : toSheafify J P ≫ η = toSheafify J P ≫ γ) : η = γ := by |
rw [sheafifyLift_unique J _ hQ _ h, ← h]
exact (sheafifyLift_unique J _ hQ _ h.symm).symm
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.SingleHomology
import Mathlib.CategoryTheory.Abelian.Homology
#align_import algebra.homology.quasi_iso from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
open CategoryTheory Limits
universe v u
variable {ι : Typ... | Mathlib/Algebra/Homology/QuasiIso.lean | 160 | 171 | theorem fromSingle₀KernelAtZeroIso_inv_eq [hf : QuasiIso' f] :
f.fromSingle₀KernelAtZeroIso.inv =
kernel.lift (X.d 0 1) (f.f 0) (by rw [f.2 0 1 rfl]; exact zero_comp) := by |
ext
dsimp only [fromSingle₀KernelAtZeroIso, CochainComplex.homology'ZeroIso, homology'OfZeroLeft,
homology'.mapIso, CochainComplex.homologyFunctor0Single₀, kernel.map]
simp only [Iso.trans_inv, Iso.app_inv, Iso.symm_inv, Category.assoc, equalizer_as_kernel,
kernel.lift_ι]
dsimp [asIso]
simp only [Cat... |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Multivariate.Basic
import Mathlib.Data.PFunctor.Multivariate.M
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.cofix from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/QPF/Multivariate/Constructions/Cofix.lean | 284 | 299 | theorem Cofix.bisim {α : TypeVec n} (r : Cofix F α → Cofix F α → Prop)
(h : ∀ x y, r x y → LiftR (RelLast α r (i := _)) (Cofix.dest x) (Cofix.dest y)) :
∀ x y, r x y → x = y := by |
apply Cofix.bisim_rel
intro x y rxy
rcases (liftR_iff (fun a b => RelLast α r a b) (dest x) (dest y)).mp (h x y rxy)
with ⟨a, f₀, f₁, dxeq, dyeq, h'⟩
rw [dxeq, dyeq, ← abs_map, ← abs_map, MvPFunctor.map_eq, MvPFunctor.map_eq]
rw [← split_dropFun_lastFun f₀, ← split_dropFun_lastFun f₁]
rw [appendFun_com... |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (K V : Type*) [Field K] [AddCommGroup V] [Module K V]
namespace Projectivization
open scoped LinearAlgebra.Projectivi... | Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 196 | 197 | theorem span_sup {S : Set (ℙ K V)} {W : Subspace K V} : span S ⊔ W = span (S ∪ W) := by |
rw [span_union, span_coe]
|
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {α : Type u} {r : α → α → Prop} {a : α} {l... | Mathlib/Data/List/Sort.lean | 145 | 148 | theorem Sorted.rel_get_of_le [IsRefl α r] {l : List α} (h : l.Sorted r) {a b : Fin l.length}
(hab : a ≤ b) : r (l.get a) (l.get b) := by |
rcases hab.eq_or_lt with (rfl | hlt)
exacts [refl _, h.rel_get_of_lt hlt]
|
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Init.Data.Bool.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Tactic.Substs
import Mathlib.Tactic.Conv
#align_import logic.equiv.defs from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
universe u... | Mathlib/Logic/Equiv/Defs.lean | 319 | 321 | theorem apply_eq_iff_eq_symm_apply {x : α} {y : β} (f : α ≃ β) : f x = y ↔ x = f.symm y := by |
conv_lhs => rw [← apply_symm_apply f y]
rw [apply_eq_iff_eq]
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 475 | 476 | theorem normSq_pos {z : K} : 0 < normSq z ↔ z ≠ 0 := by |
rw [lt_iff_le_and_ne, Ne, eq_comm]; simp [normSq_nonneg]
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 296 | 303 | theorem ae_measure_preimage_mul_right_lt_top_of_ne_zero [IsMulLeftInvariant ν]
(sm : MeasurableSet s) (h2s : ν s ≠ 0) (h3s : ν s ≠ ∞) :
∀ᵐ x ∂μ, ν ((fun y => y * x) ⁻¹' s) < ∞ := by |
refine (ae_measure_preimage_mul_right_lt_top ν ν sm h3s).filter_mono ?_
refine (absolutelyContinuous_of_isMulLeftInvariant μ ν ?_).ae_le
refine mt ?_ h2s
intro hν
rw [hν, Measure.coe_zero, Pi.zero_apply]
|
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