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.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 174 | 175 | theorem linearEquiv_det_rotation (a : circle) : LinearEquiv.det (rotation a).toLinearEquiv = 1 := by |
rw [← Units.eq_iff, LinearEquiv.coe_det, det_rotation, Units.val_one]
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 363 | 365 | theorem ball_add_closedBall (hε : 0 < ε) (hδ : 0 ≤ δ) (a b : E) :
ball a ε + closedBall b δ = ball (a + b) (ε + δ) := by |
rw [ball_add, thickening_closedBall hε hδ b, Metric.vadd_ball, vadd_eq_add]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 95 | 98 | theorem lt_natDegree_of_mem_eraseLead_support {a : ℕ} (h : a ∈ (eraseLead f).support) :
a < f.natDegree := by |
rw [eraseLead_support, mem_erase] at h
exact (le_natDegree_of_mem_supp a h.2).lt_of_ne h.1
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 462 | 464 | theorem equicontinuousAt_iff_range {F : ι → X → α} {x₀ : X} :
EquicontinuousAt F x₀ ↔ EquicontinuousAt ((↑) : range F → X → α) x₀ := by |
simp only [EquicontinuousAt, forall_subtype_range_iff]
|
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 162 | 164 | theorem _root_.AlgHom.map_algebraMap (f : A →ₐ[S] B) (r : R) :
f (algebraMap R A r) = algebraMap R B r := by |
rw [algebraMap_apply R S A r, f.commutes, ← algebraMap_apply R S B]
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 602 | 620 | theorem MonotoneOn.eVariationOn_le {f : α → ℝ} {s : Set α} (hf : MonotoneOn f s) {a b : α}
(as : a ∈ s) (bs : b ∈ s) : eVariationOn f (s ∩ Icc a b) ≤ ENNReal.ofReal (f b - f a) := by |
apply iSup_le _
rintro ⟨n, ⟨u, hu, us⟩⟩
calc
(∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i))) =
∑ i ∈ Finset.range n, ENNReal.ofReal (f (u (i + 1)) - f (u i)) := by
refine Finset.sum_congr rfl fun i hi => ?_
simp only [Finset.mem_range] at hi
rw [edist_dist, Real.dist_eq, abs... |
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 633 | 636 | theorem sum_fiberwise {α M} [AddCommMonoid M] (π : Prepartition I) (f : Box ι → α) (g : Box ι → M) :
(∑ y ∈ π.boxes.image f, ∑ J ∈ (π.filter fun J => f J = y).boxes, g J) =
∑ J ∈ π.boxes, g J := by |
convert sum_fiberwise_of_maps_to (fun _ => Finset.mem_image_of_mem f) g
|
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Pointwise
import Mathlib.Data.Real.Archimedean
#align_import data.real.pointwise from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
open Set
open Pointwise
variable {ι : Sort*} {α : Type*} [LinearOrde... | Mathlib/Data/Real/Pointwise.lean | 91 | 100 | theorem Real.sSup_smul_of_nonpos (ha : a ≤ 0) (s : Set ℝ) : sSup (a • s) = a • sInf s := by |
obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sSup... |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 252 | 255 | theorem _root_.fourierIntegral_gaussian_pi (hb : 0 < b.re) :
(𝓕 fun (x : ℝ) ↦ cexp (-π * b * x ^ 2)) =
fun t : ℝ ↦ 1 / b ^ (1 / 2 : ℂ) * cexp (-π / b * t ^ 2) := by |
simpa only [mul_zero, zero_mul, add_zero] using fourierIntegral_gaussian_pi' hb 0
|
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 98 | 106 | theorem not_mem_iff (p : T.CompleteType α) (φ : L[[α]].Sentence) : φ.not ∈ p ↔ ¬φ ∈ p :=
⟨fun hf ht => by
have h : ¬IsSatisfiable ({φ, φ.not} : L[[α]].Theory) := by |
rintro ⟨@⟨_, _, h, _⟩⟩
simp only [model_iff, mem_insert_iff, mem_singleton_iff, forall_eq_or_imp, forall_eq] at h
exact h.2 h.1
refine h (p.isMaximal.1.mono ?_)
rw [insert_subset_iff, singleton_subset_iff]
exact ⟨ht, hf⟩, (p.mem_or_not_mem φ).resolve_left⟩
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.TFAE
import Mathlib.Topology.Order.Monotone
#align_import set_theory.ordinal.topology from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
universe u v
open Cardinal Order Topology
namespace Ordina... | Mathlib/SetTheory/Ordinal/Topology.lean | 132 | 135 | theorem mem_closed_iff_sup (hs : IsClosed s) :
a ∈ s ↔ ∃ (ι : Type u) (_hι : Nonempty ι) (f : ι → Ordinal),
(∀ i, f i ∈ s) ∧ sup.{u, u} f = a := by |
rw [← mem_closure_iff_sup, hs.closure_eq]
|
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 198 | 206 | theorem set_lintegral_preimage_condDistrib (hX : Measurable X) (hY : AEMeasurable Y μ)
(hs : MeasurableSet s) (ht : MeasurableSet t) :
∫⁻ a in X ⁻¹' t, condDistrib Y X μ (X a) s ∂μ = μ (X ⁻¹' t ∩ Y ⁻¹' s) := by |
-- Porting note: need to massage the LHS integrand into the form accepted by `lintegral_comp`
-- (`rw` does not see that the two forms are defeq)
conv_lhs => arg 2; change (fun a => ((condDistrib Y X μ) a) s) ∘ X
rw [lintegral_comp (kernel.measurable_coe _ hs) hX, condDistrib, ← Measure.restrict_map hX ht, ←
... |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 95 | 99 | theorem gauge_zero : gauge s 0 = 0 := by |
rw [gauge_def']
by_cases h : (0 : E) ∈ s
· simp only [smul_zero, sep_true, h, csInf_Ioi]
· simp only [smul_zero, sep_false, h, Real.sInf_empty]
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
noncomputable section
open Set hiding restrict restrict_apply
open Filter ENNReal
open Function (support)
open scoped Cla... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 180 | 189 | theorem measurableSet_cut (r : α → β → Prop) (f : α →ₛ β) (h : ∀ b, MeasurableSet { a | r a b }) :
MeasurableSet { a | r a (f a) } := by |
have : { a | r a (f a) } = ⋃ b ∈ range f, { a | r a b } ∩ f ⁻¹' {b} := by
ext a
suffices r a (f a) ↔ ∃ i, r a (f i) ∧ f a = f i by simpa
exact ⟨fun h => ⟨a, ⟨h, rfl⟩⟩, fun ⟨a', ⟨h', e⟩⟩ => e.symm ▸ h'⟩
rw [this]
exact
MeasurableSet.biUnion f.finite_range.countable fun b _ =>
MeasurableSet.i... |
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 211 | 212 | theorem inv_hom_whiskerRight {f g : a ⟶ b} (η : f ≅ g) (h : b ⟶ c) :
η.inv ▷ h ≫ η.hom ▷ h = 𝟙 (g ≫ h) := by | rw [← comp_whiskerRight, inv_hom_id, id_whiskerRight]
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Algebra.SMulWithZero
import Mathlib.Order.Hom.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import algebra.tropical.basic from "leanprover-community/mathlib"@"9116dd6709f303dcf781632e15fdef382b0fc579"
... | Mathlib/Algebra/Tropical/Basic.lean | 357 | 359 | theorem add_eq_iff {x y z : Tropical R} : x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x := by |
rw [trop_add_def, trop_eq_iff_eq_untrop]
simp [min_eq_iff]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.Polynomial.Nilpotent
import Mathlib.RingTheory.Polynomial.Tower
open Set Function
noncomputable section
namespace Polynomial
variable {R S : Type*} [CommR... | Mathlib/Dynamics/Newton.lean | 81 | 99 | theorem aeval_pow_two_pow_dvd_aeval_iterate_newtonMap
(h : IsNilpotent (aeval x P)) (h' : IsUnit (aeval x <| derivative P)) (n : ℕ) :
(aeval x P) ^ (2 ^ n) ∣ aeval (P.newtonMap^[n] x) P := by |
induction n with
| zero => simp
| succ n ih =>
have ⟨d, hd⟩ := binomExpansion (P.map (algebraMap R S)) (P.newtonMap^[n] x)
(-Ring.inverse (aeval (P.newtonMap^[n] x) <| derivative P) * aeval (P.newtonMap^[n] x) P)
rw [eval_map_algebraMap, eval_map_algebraMap] at hd
rw [iterate_succ', comp_apply,... |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 106 | 108 | theorem const.get_map (f : α ⟹ β) (x : const n A α) : const.get (f <$$> x) = const.get x := by |
cases x
rfl
|
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.Init.Align
import Mathlib.Tactic.GCongr
import Mathlib.Tactic... | Mathlib/Algebra/Order/CauSeq/Basic.lean | 102 | 107 | theorem cauchy₂ (hf : IsCauSeq abv f) {ε : α} (ε0 : 0 < ε) :
∃ i, ∀ j ≥ i, ∀ k ≥ i, abv (f j - f k) < ε := by |
refine (hf _ (half_pos ε0)).imp fun i hi j ij k ik => ?_
rw [← add_halves ε]
refine lt_of_le_of_lt (abv_sub_le abv _ _ _) (add_lt_add (hi _ ij) ?_)
rw [abv_sub abv]; exact hi _ ik
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 327 | 333 | theorem apply_le_nfpBFamily (ho : o ≠ 0) (H : ∀ i hi, IsNormal (f i hi)) {a b} :
(∀ i hi, f i hi b ≤ nfpBFamily.{u, v} o f a) ↔ b ≤ nfpBFamily.{u, v} o f a := by |
refine ⟨fun h => ?_, fun h i hi => ?_⟩
· have ho' : 0 < o := Ordinal.pos_iff_ne_zero.2 ho
exact ((H 0 ho').self_le b).trans (h 0 ho')
· rw [← nfpBFamily_fp (H i hi)]
exact (H i hi).monotone h
|
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.ideal.associated_prime from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [... | Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 172 | 183 | theorem associatedPrimes.eq_singleton_of_isPrimary [IsNoetherianRing R] (hI : I.IsPrimary) :
associatedPrimes R (R ⧸ I) = {I.radical} := by |
ext J
rw [Set.mem_singleton_iff]
refine ⟨IsAssociatedPrime.eq_radical hI, ?_⟩
rintro rfl
haveI : Nontrivial (R ⧸ I) := by
refine ⟨(Ideal.Quotient.mk I : _) 1, (Ideal.Quotient.mk I : _) 0, ?_⟩
rw [Ne, Ideal.Quotient.eq, sub_zero, ← Ideal.eq_top_iff_one]
exact hI.1
obtain ⟨a, ha⟩ := associatedPri... |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v v₂ u u' u₂
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 91 | 92 | theorem KernelFork.app_one (s : KernelFork f) : s.π.app one = 0 := by |
simp [Fork.app_one_eq_ι_comp_right]
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 478 | 479 | theorem convexHull_sub (s t : Set E) : convexHull R (s - t) = convexHull R s - convexHull R t := by |
simp_rw [sub_eq_add_neg, convexHull_add, ← convexHull_neg]
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 134 | 144 | theorem sum_incMatrix_apply_of_mem_edgeSet [Fintype α] :
e ∈ G.edgeSet → ∑ a, G.incMatrix R a e = 2 := by |
classical
refine e.ind ?_
intro a b h
rw [mem_edgeSet] at h
rw [← Nat.cast_two, ← card_pair h.ne]
simp only [incMatrix_apply', sum_boole, mk'_mem_incidenceSet_iff, h, true_and_iff]
congr 2
ext e
simp only [mem_filter, mem_univ, true_and_iff, mem_insert, mem_singleton]
|
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.projective_resolution from "leanprover-community/mathlib"@"324a7502510e835cdbd3de1519b6c66b51fb2467"
universe v u
namespace CategoryTheory
open Category Limits ChainComplex HomologicalComplex
variable {C : Type u} [Category.{v} ... | Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean | 102 | 104 | theorem complex_d_succ_comp (n : ℕ) :
P.complex.d n (n + 1) ≫ P.complex.d (n + 1) (n + 2) = 0 := by |
simp
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α... | Mathlib/Algebra/BigOperators/Pi.lean | 69 | 72 | theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by |
ext
simp
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 331 | 333 | theorem Antivary.sum_smul_lt_sum_comp_perm_smul_iff (hfg : Antivary f g) :
((∑ i, f i • g i) < ∑ i, f (σ i) • g i) ↔ ¬Antivary (f ∘ σ) g := by |
simp [(hfg.antivaryOn _).sum_smul_lt_sum_comp_perm_smul_iff fun _ _ ↦ mem_univ _]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.convergence_in_measure from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open TopologicalSpace Filter
ope... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 300 | 309 | theorem tendstoInMeasure_of_tendsto_snorm_of_ne_top (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : ∀ n, AEStronglyMeasurable (f n) μ) (hg : AEStronglyMeasurable g μ) {l : Filter ι}
(hfg : Tendsto (fun n => snorm (f n - g) p μ) l (𝓝 0)) : TendstoInMeasure μ f l g := by |
refine TendstoInMeasure.congr (fun i => (hf i).ae_eq_mk.symm) hg.ae_eq_mk.symm ?_
refine tendstoInMeasure_of_tendsto_snorm_of_stronglyMeasurable
hp_ne_zero hp_ne_top (fun i => (hf i).stronglyMeasurable_mk) hg.stronglyMeasurable_mk ?_
have : (fun n => snorm ((hf n).mk (f n) - hg.mk g) p μ) = fun n => snorm (f... |
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 | 578 | 582 | theorem kahler_ne_zero_iff (x y : E) : o.kahler x y ≠ 0 ↔ x ≠ 0 ∧ y ≠ 0 := by |
refine ⟨?_, fun h => o.kahler_ne_zero h.1 h.2⟩
contrapose
simp only [not_and_or, Classical.not_not, kahler_apply_apply, Complex.real_smul]
rintro (rfl | rfl) <;> simp
|
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 259 | 266 | theorem injective_mapNatBool [MeasurableSpace α] [CountablyGenerated α]
[SeparatesPoints α] : Injective (mapNatBool α) := by |
intro x y hxy
rw [← generateFrom_natGeneratingSequence α] at *
apply separating_of_generateFrom (range (natGeneratingSequence _))
rintro - ⟨n, rfl⟩
rw [← decide_eq_decide]
exact congr_fun hxy n
|
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 | 573 | 575 | theorem kahler_ne_zero {x y : E} (hx : x ≠ 0) (hy : y ≠ 0) : o.kahler x y ≠ 0 := by |
apply mt o.eq_zero_or_eq_zero_of_kahler_eq_zero
tauto
|
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
noncomputable section
open Filter Finset Function Encodable
open scoped Topology
variable {M : Type*} [CommMonoid M] [TopologicalSpace M] {m m' : M}
variable {G : Type*} [CommGroup G] {g g' : G}
-- don't declare [Topologic... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 73 | 78 | theorem even_mul_odd {f : ℕ → M} (he : HasProd (fun k ↦ f (2 * k)) m)
(ho : HasProd (fun k ↦ f (2 * k + 1)) m') : HasProd f (m * m') := by |
have := mul_right_injective₀ (two_ne_zero' ℕ)
replace ho := ((add_left_injective 1).comp this).hasProd_range_iff.2 ho
refine (this.hasProd_range_iff.2 he).mul_isCompl ?_ ho
simpa [(· ∘ ·)] using Nat.isCompl_even_odd
|
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open F... | Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 89 | 90 | theorem hasCompactSupport_normed : HasCompactSupport (f.normed μ) := by |
simp only [HasCompactSupport, f.tsupport_normed_eq (μ := μ), isCompact_closedBall]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 292 | 295 | theorem tan_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
Real.tan (angle x (x - y)) = ‖y‖ / ‖x‖ := by |
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg, tan_angle_add_of_inner_eq_zero h, norm_neg]
|
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 60 | 70 | theorem powerset_aux'_perm {l₁ l₂ : List α} (p : l₁ ~ l₂) : powersetAux' l₁ ~ powersetAux' l₂ := by |
induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ _ _ IH₁ IH₂
· simp
· simp only [powersetAux'_cons]
exact IH.append (IH.map _)
· simp only [powersetAux'_cons, map_append, List.map_map, append_assoc]
apply Perm.append_left
rw [← append_assoc, ← append_assoc,
(by funext s; simp [cons_swap] : cons ... |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 58 | 61 | theorem ghostComponent_zero_verschiebungFun (x : 𝕎 R) :
ghostComponent 0 (verschiebungFun x) = 0 := by |
rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 336 | 339 | theorem dual_rotateL (l : Ordnode α) (x : α) (r : Ordnode α) :
dual (rotateL l x r) = rotateR (dual r) x (dual l) := by |
cases r <;> simp [rotateL, rotateR, dual_node']; split_ifs <;>
simp [dual_node3L, dual_node4L, node3R, add_comm]
|
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 205 | 210 | theorem contractLeft_contractLeft (x : CliffordAlgebra Q) : d⌋(d⌋x) = 0 := by |
induction' x using CliffordAlgebra.left_induction with r x y hx hy m x hx
· simp_rw [contractLeft_algebraMap, map_zero]
· rw [map_add, map_add, hx, hy, add_zero]
· rw [contractLeft_ι_mul, map_sub, contractLeft_ι_mul, hx, LinearMap.map_smul,
mul_zero, sub_zero, sub_self]
|
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 24 | 25 | theorem zero_right [MulZeroClass G₀] (a : G₀) : SemiconjBy a 0 0 := by |
simp only [SemiconjBy, mul_zero, zero_mul]
|
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 39 | 40 | theorem toFinset_none [Decidable (none : Part α).Dom] : none.toFinset = (∅ : Finset α) := by |
simp [toFinset]
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 119 | 121 | theorem rotate_cons_succ (l : List α) (a : α) (n : ℕ) :
(a :: l : List α).rotate (n + 1) = (l ++ [a]).rotate n := by |
rw [rotate_eq_rotate', rotate_eq_rotate', rotate'_cons_succ]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 531 | 532 | theorem zero_isRoot_of_coeff_zero_eq_zero {p : R[X]} (hp : p.coeff 0 = 0) : IsRoot p 0 := by |
rwa [coeff_zero_eq_eval_zero] at hp
|
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 228 | 230 | theorem contractRight_comm (x : CliffordAlgebra Q) : x⌊d⌊d' = -(x⌊d'⌊d) := by |
rw [contractRight_eq, contractRight_eq, contractRight_eq, contractRight_eq, reverse_reverse,
reverse_reverse, contractLeft_comm, map_neg]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 315 | 316 | theorem nthRoots_zero (r : R) : nthRoots 0 r = 0 := by |
simp only [empty_eq_zero, pow_zero, nthRoots, ← C_1, ← C_sub, roots_C]
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,646 | 1,649 | theorem isOpen_sigma_fst_preimage (s : Set ι) : IsOpen (Sigma.fst ⁻¹' s : Set (Σ a, σ a)) := by |
rw [← biUnion_of_singleton s, preimage_iUnion₂]
simp only [← range_sigmaMk]
exact isOpen_biUnion fun _ _ => isOpen_range_sigmaMk
|
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 81 | 85 | theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
IsNilpotent (f r) := by |
use hr.choose
rw [← map_pow, hr.choose_spec, map_zero]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Interval.Set.IsoIoo
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.UrysohnsBounded
#align_import topology.tietze_extension from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {X Y : T... | Mathlib/Topology/TietzeExtension.lean | 269 | 272 | theorem exists_extension_norm_eq_of_closedEmbedding (f : X →ᵇ ℝ) {e : X → Y}
(he : ClosedEmbedding e) : ∃ g : Y →ᵇ ℝ, ‖g‖ = ‖f‖ ∧ g ∘ e = f := by |
rcases exists_extension_norm_eq_of_closedEmbedding' f ⟨e, he.continuous⟩ he with ⟨g, hg, rfl⟩
exact ⟨g, hg, rfl⟩
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | theorem mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s := by |
apply iSup_le _
rintro ⟨n, ⟨u, hu, ut⟩⟩
exact sum_le f n hu fun i => hst (ut i)
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 83 | 85 | theorem Equiv.pointReflection_midpoint_right (x y : P) :
(Equiv.pointReflection (midpoint R x y)) y = x := by |
rw [midpoint_comm, Equiv.pointReflection_midpoint_left]
|
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 119 | 120 | theorem cast_mk (a b : ℤ) : (a /. b : α) = a / b := by |
simp only [divInt_eq_div, cast_div, cast_intCast]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 116 | 119 | theorem angle_add_le_pi_div_two_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
angle x (x + y) ≤ π / 2 := by |
rw [angle_add_eq_arccos_of_inner_eq_zero h, Real.arccos_le_pi_div_two]
exact div_nonneg (norm_nonneg _) (norm_nonneg _)
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 119 | 122 | theorem comp_left_id (r : Rel α β) : @Eq α • r = r := by |
unfold comp
ext x
simp
|
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 178 | 180 | theorem integral_truncation_eq_intervalIntegral (hf : AEStronglyMeasurable f μ) {A : ℝ}
(hA : 0 ≤ A) : ∫ x, truncation f A x ∂μ = ∫ y in -A..A, y ∂Measure.map f μ := by |
simpa using moment_truncation_eq_intervalIntegral hf hA one_ne_zero
|
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 466 | 472 | theorem IsClique.card_le_of_coloring {s : Finset V} (h : G.IsClique s) [Fintype α]
(C : G.Coloring α) : s.card ≤ Fintype.card α := by |
rw [isClique_iff_induce_eq] at h
have f : G.induce ↑s ↪g G := Embedding.comap (Function.Embedding.subtype fun x => x ∈ ↑s) G
rw [h] at f
convert Fintype.card_le_of_injective _ (C.comp f.toHom).injective_of_top_hom using 1
simp
|
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Topology.Algebra.Module.FiniteDimension
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.normed_space.hahn_banach.separation from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a... | Mathlib/Analysis/NormedSpace/HahnBanach/Separation.lean | 207 | 214 | theorem iInter_halfspaces_eq (hs₁ : Convex ℝ s) (hs₂ : IsClosed s) :
⋂ l : E →L[ℝ] ℝ, { x | ∃ y ∈ s, l x ≤ l y } = s := by |
rw [Set.iInter_setOf]
refine Set.Subset.antisymm (fun x hx => ?_) fun x hx l => ⟨x, hx, le_rfl⟩
by_contra h
obtain ⟨l, s, hlA, hl⟩ := geometric_hahn_banach_closed_point hs₁ hs₂ h
obtain ⟨y, hy, hxy⟩ := hx l
exact ((hxy.trans_lt (hlA y hy)).trans hl).not_le le_rfl
|
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 159 | 166 | theorem openEmbedding_iff_openEmbedding_of_iSup_eq_top (h : Continuous f) :
OpenEmbedding f ↔ ∀ i, OpenEmbedding ((U i).1.restrictPreimage f) := by |
simp_rw [openEmbedding_iff]
rw [forall_and]
apply and_congr
· apply embedding_iff_embedding_of_iSup_eq_top <;> assumption
· simp_rw [Set.range_restrictPreimage]
apply isOpen_iff_coe_preimage_of_iSup_eq_top hU
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
variable {α : Type*} [DecidableEq α] {m : Multiset α}
def Multiset.ToType (m : Multiset α) : Type _ := (x : α) × Fi... | Mathlib/Data/Multiset/Fintype.lean | 248 | 251 | theorem Multiset.prod_eq_prod_coe [CommMonoid α] (m : Multiset α) : m.prod = ∏ x : m, (x : α) := by |
congr
-- Porting note: `simp` fails with "maximum recursion depth has been reached"
erw [map_univ_coe]
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 55 | 56 | theorem hermite_succ (n : ℕ) : hermite (n + 1) = X * hermite n - derivative (hermite n) := by |
rw [hermite]
|
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 93 | 100 | theorem rotate'_length_mul (l : List α) : ∀ n : ℕ, l.rotate' (l.length * n) = l
| 0 => by simp
| n + 1 =>
calc
l.rotate' (l.length * (n + 1)) =
(l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by |
simp [-rotate'_length, Nat.mul_succ, rotate'_rotate']
_ = l := by rw [rotate'_length, rotate'_length_mul l n]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 569 | 570 | theorem arg_coe_angle_eq_iff {x y : ℂ} : (arg x : Real.Angle) = arg y ↔ arg x = arg y := by |
simp_rw [← Real.Angle.toReal_inj, arg_coe_angle_toReal_eq_arg]
|
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 | 1,039 | 1,039 | theorem zero_mod (b : Ordinal) : 0 % b = 0 := by | simp only [mod_def, zero_div, mul_zero, sub_self]
|
import Mathlib.Data.Matroid.Basic
open Set Matroid
variable {α : Type*} {I B X : Set α}
section IndepMatroid
structure IndepMatroid (α : Type*) where
(E : Set α)
(Indep : Set α → Prop)
(indep_empty : Indep ∅)
(indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
(indep_aug : ∀⦃I B⦄, Indep I → I ∉ ... | Mathlib/Data/Matroid/IndepAxioms.lean | 265 | 281 | theorem _root_.Matroid.existsMaximalSubsetProperty_of_bdd {P : Set α → Prop}
(hP : ∃ (n : ℕ), ∀ Y, P Y → Y.encard ≤ n) (X : Set α) : ExistsMaximalSubsetProperty P X := by |
obtain ⟨n, hP⟩ := hP
rintro I hI hIX
have hfin : Set.Finite (ncard '' {Y | P Y ∧ I ⊆ Y ∧ Y ⊆ X}) := by
rw [finite_iff_bddAbove, bddAbove_def]
simp_rw [ENat.le_coe_iff] at hP
use n
rintro x ⟨Y, ⟨hY,-,-⟩, rfl⟩
obtain ⟨n₀, heq, hle⟩ := hP Y hY
rwa [ncard_def, heq, ENat.toNat_coe]
-- have... |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv 𝕜 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
· rw [iteratedFDeriv_zero_apply, add_zero]
· rw [tsum_eq_single 0 fun n hn ↦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 331 | 339 | theorem descPochhammer_succ_comp_X_sub_one (n : ℕ) :
(descPochhammer R (n + 1)).comp (X - 1) =
descPochhammer R (n + 1) - (n + (1 : R[X])) • (descPochhammer R n).comp (X - 1) := by |
suffices (descPochhammer ℤ (n + 1)).comp (X - 1) =
descPochhammer ℤ (n + 1) - (n + 1) * (descPochhammer ℤ n).comp (X - 1)
by simpa [map_comp] using congr_arg (Polynomial.map (Int.castRingHom R)) this
nth_rw 2 [descPochhammer_succ_left]
rw [← sub_mul, descPochhammer_succ_right ℤ n, mul_comp, mul_comm, s... |
import Mathlib.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@"431589bce478b2229eba14b14a283250428217db"
noncomputable section
open scoped Classical
open Manifold Topology
open Set Filter TopologicalSpace
variable {H M H' M' X : Typ... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 121 | 136 | theorem left_invariance {s : Set H} {x : H} {f : H → H'} {e' : PartialHomeomorph H' H'}
(he' : e' ∈ G') (hfs : ContinuousWithinAt f s x) (hxe' : f x ∈ e'.source) :
P (e' ∘ f) s x ↔ P f s x := by |
have h2f := hfs.preimage_mem_nhdsWithin (e'.open_source.mem_nhds hxe')
have h3f :=
((e'.continuousAt hxe').comp_continuousWithinAt hfs).preimage_mem_nhdsWithin <|
e'.symm.open_source.mem_nhds <| e'.mapsTo hxe'
constructor
· intro h
rw [hG.is_local_nhds h3f] at h
have h2 := hG.left_invariance'... |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 156 | 165 | theorem exp_eq_one_iff {x : ℂ} : exp x = 1 ↔ ∃ n : ℤ, x = n * (2 * π * I) := by |
constructor
· intro h
rcases existsUnique_add_zsmul_mem_Ioc Real.two_pi_pos x.im (-π) with ⟨n, hn, -⟩
use -n
rw [Int.cast_neg, neg_mul, eq_neg_iff_add_eq_zero]
have : (x + n * (2 * π * I)).im ∈ Set.Ioc (-π) π := by simpa [two_mul, mul_add] using hn
rw [← log_exp this.1 this.2, exp_periodic.int_... |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 60 | 65 | theorem TensorProduct.isTensorProduct : IsTensorProduct (TensorProduct.mk R M N) := by |
delta IsTensorProduct
convert_to Function.Bijective (LinearMap.id : M ⊗[R] N →ₗ[R] M ⊗[R] N) using 2
· apply TensorProduct.ext'
simp
· exact Function.bijective_id
|
import Mathlib.Logic.Nonempty
import Mathlib.Init.Set
import Mathlib.Logic.Basic
#align_import logic.function.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Function
universe u v w
namespace Function
section
variable {α β γ : Sort*} {f : α → β}
@[reducible, simp] de... | Mathlib/Logic/Function/Basic.lean | 89 | 91 | theorem Injective.beq_eq {α β : Type*} [BEq α] [LawfulBEq α] [BEq β] [LawfulBEq β] {f : α → β}
(I : Injective f) {a b : α} : (f a == f b) = (a == b) := by |
by_cases h : a == b <;> simp [h] <;> simpa [I.eq_iff] using h
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 1,225 | 1,225 | theorem abs_zero : (0 : EReal).abs = 0 := by | rw [abs_eq_zero_iff]
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 404 | 424 | theorem exists_norm_le_le_norm_sub_of_finset {c : 𝕜} (hc : 1 < ‖c‖) {R : ℝ} (hR : ‖c‖ < R)
(h : ¬FiniteDimensional 𝕜 E) (s : Finset E) : ∃ x : E, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖ := by |
let F := Submodule.span 𝕜 (s : Set E)
haveI : FiniteDimensional 𝕜 F :=
Module.finite_def.2
((Submodule.fg_top _).2 (Submodule.fg_def.2 ⟨s, Finset.finite_toSet _, rfl⟩))
have Fclosed : IsClosed (F : Set E) := Submodule.closed_of_finiteDimensional _
have : ∃ x, x ∉ F := by
contrapose! h
have ... |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 513 | 515 | theorem lt_add_one {x : PartENat} (hx : x ≠ ⊤) : x < x + 1 := by |
rw [PartENat.lt_add_iff_pos_right hx]
norm_cast
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 303 | 305 | theorem div_const {𝕜 : Type*} {f : ℝ → 𝕜} [NormedField 𝕜] (h : IntervalIntegrable f μ a b)
(c : 𝕜) : IntervalIntegrable (fun x => f x / c) μ a b := by |
simpa only [div_eq_mul_inv] using mul_const h c⁻¹
|
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 326 | 339 | theorem ae_eq_const_or_norm_average_lt_of_norm_le_const [StrictConvexSpace ℝ E]
(h_le : ∀ᵐ x ∂μ, ‖f x‖ ≤ C) : f =ᵐ[μ] const α (⨍ x, f x ∂μ) ∨ ‖⨍ x, f x ∂μ‖ < C := by |
rcases le_or_lt C 0 with hC0 | hC0
· have : f =ᵐ[μ] 0 := h_le.mono fun x hx => norm_le_zero_iff.1 (hx.trans hC0)
simp only [average_congr this, Pi.zero_apply, average_zero]
exact Or.inl this
by_cases hfi : Integrable f μ; swap
· simp [average_eq, integral_undef hfi, hC0, ENNReal.toReal_pos_iff]
rcase... |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 343 | 353 | theorem map_nullHomotopicMap' {W : Type*} [Category W] [Preadditive W] (G : V ⥤ W) [G.Additive]
(hom : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(G.mapHomologicalComplex c).map (nullHomotopicMap' hom) =
nullHomotopicMap' fun i j hij => by exact G.map (hom i j hij) := by |
ext n
erw [map_nullHomotopicMap]
congr
ext i j
split_ifs
· rfl
· rw [G.map_zero]
|
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 | 637 | 639 | theorem neg_pi_div_two_ne_zero : ((-π / 2 : ℝ) : Angle) ≠ 0 := by |
rw [← toReal_injective.ne_iff, toReal_neg_pi_div_two, toReal_zero]
exact div_ne_zero (neg_ne_zero.2 Real.pi_ne_zero) two_ne_zero
|
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 283 | 288 | theorem chromaticNumber_ne_top_iff_exists : G.chromaticNumber ≠ ⊤ ↔ ∃ n, G.Colorable n := by |
rw [chromaticNumber]
convert_to ⨅ n : {m | G.Colorable m}, (n : ℕ∞) ≠ ⊤ ↔ _
· rw [iInf_subtype]
rw [← lt_top_iff_ne_top, ENat.iInf_coe_lt_top]
simp
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.GroupAction.Opposite
#align_import ring_theory.non_zero_divisors from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
variable (M₀ : Type*) [... | Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 182 | 191 | theorem mul_mem_nonZeroDivisors {a b : M₁} : a * b ∈ M₁⁰ ↔ a ∈ M₁⁰ ∧ b ∈ M₁⁰ := by |
constructor
· intro h
constructor <;> intro x h' <;> apply h
· rw [← mul_assoc, h', zero_mul]
· rw [mul_comm a b, ← mul_assoc, h', zero_mul]
· rintro ⟨ha, hb⟩ x hx
apply ha
apply hb
rw [mul_assoc, hx]
|
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 | 252 | 269 | theorem continuousAt_gaussian_integral (b : ℂ) (hb : 0 < re b) :
ContinuousAt (fun c : ℂ => ∫ x : ℝ, cexp (-c * (x : ℂ) ^ 2)) b := by |
let f : ℂ → ℝ → ℂ := fun (c : ℂ) (x : ℝ) => cexp (-c * (x : ℂ) ^ 2)
obtain ⟨d, hd, hd'⟩ := exists_between hb
have f_meas : ∀ c : ℂ, AEStronglyMeasurable (f c) volume := fun c => by
apply Continuous.aestronglyMeasurable
exact Complex.continuous_exp.comp (continuous_const.mul (continuous_ofReal.pow 2))
h... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 153 | 161 | theorem eval₂_sum (p : T[X]) (g : ℕ → T → R[X]) (x : S) :
(p.sum g).eval₂ f x = p.sum fun n a => (g n a).eval₂ f x := by |
let T : R[X] →+ S :=
{ toFun := eval₂ f x
map_zero' := eval₂_zero _ _
map_add' := fun p q => eval₂_add _ _ }
have A : ∀ y, eval₂ f x y = T y := fun y => rfl
simp only [A]
rw [sum, map_sum, sum]
|
import Mathlib.Analysis.Fourier.Inversion
open Real Complex Set MeasureTheory
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
open scoped FourierTransform
private theorem rexp_neg_deriv_aux :
∀ x ∈ univ, HasDerivWithinAt (rexp ∘ Neg.neg) (-rexp (-x)) univ x :=
fun x _ ↦ mul_neg_one (rexp (-x)... | Mathlib/Analysis/MellinInversion.lean | 89 | 121 | theorem mellin_inversion (σ : ℝ) (f : ℝ → E) {x : ℝ} (hx : 0 < x) (hf : MellinConvergent f σ)
(hFf : VerticalIntegrable (mellin f) σ) (hfx : ContinuousAt f x) :
mellinInv σ (mellin f) x = f x := by |
let g := fun (u : ℝ) => Real.exp (-σ * u) • f (Real.exp (-u))
replace hf : Integrable g := by
rw [MellinConvergent, ← rexp_neg_image_aux, integrableOn_image_iff_integrableOn_abs_deriv_smul
MeasurableSet.univ rexp_neg_deriv_aux rexp_neg_injOn_aux] at hf
replace hf : Integrable fun (x : ℝ) ↦ cexp (-↑σ ... |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 40 | 41 | theorem gcd_add_mul_left_right (m n k : ℕ) : gcd m (n + m * k) = gcd m n := by |
simp [gcd_rec m (n + m * k), gcd_rec m n]
|
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Order.OrderIsoNat
#align_import data.nat.nth from "leanprover-community/mathlib"@"7fdd4f3746cb059edfdb5d52cba98f66fce418c0"
open Finset
namespace Nat
variable (p : ℕ → Prop)
noncomputable d... | Mathlib/Data/Nat/Nth.lean | 354 | 355 | theorem count_nth_succ {n : ℕ} (hn : ∀ hf : (setOf p).Finite, n < hf.toFinset.card) :
count p (nth p n + 1) = n + 1 := by | rw [count_succ, count_nth hn, if_pos (nth_mem _ hn)]
|
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 125 | 128 | theorem neg_one_pow_div_two_of_three_mod_four {n : ℕ} (hn : n % 4 = 3) :
(-1 : ℤ) ^ (n / 2) = -1 := by |
rw [← χ₄_eq_neg_one_pow (Nat.odd_of_mod_four_eq_three hn), ← natCast_mod, hn]
rfl
|
import Mathlib.Logic.Encodable.Basic
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.Subsingleton
#align_import logic.encodable.lattice from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
namespace Encodable
variable {α : Type*} {β : Type*} [Encodable β]
| Mathlib/Logic/Encodable/Lattice.lean | 30 | 33 | theorem iSup_decode₂ [CompleteLattice α] (f : β → α) :
⨆ (i : ℕ) (b ∈ decode₂ β i), f b = (⨆ b, f b) := by |
rw [iSup_comm]
simp only [mem_decode₂, iSup_iSup_eq_right]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Tactic.ComputeDegree
#align_import data.polynomial.cancel_leads from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace Polynomial
noncomputable section
open Polyn... | Mathlib/Algebra/Polynomial/CancelLeads.lean | 52 | 71 | theorem natDegree_cancelLeads_lt_of_natDegree_le_natDegree_of_comm
(comm : p.leadingCoeff * q.leadingCoeff = q.leadingCoeff * p.leadingCoeff)
(h : p.natDegree ≤ q.natDegree) (hq : 0 < q.natDegree) :
(p.cancelLeads q).natDegree < q.natDegree := by |
by_cases hp : p = 0
· convert hq
simp [hp, cancelLeads]
rw [cancelLeads, sub_eq_add_neg, tsub_eq_zero_iff_le.mpr h, pow_zero, mul_one]
by_cases h0 :
C p.leadingCoeff * q + -(C q.leadingCoeff * X ^ (q.natDegree - p.natDegree) * p) = 0
· exact (le_of_eq (by simp only [h0, natDegree_zero])).trans_lt hq
... |
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 | 628 | 629 | theorem toReal_eq_neg_pi_div_two_iff {θ : Angle} : θ.toReal = -π / 2 ↔ θ = (-π / 2 : ℝ) := by |
rw [← toReal_inj, toReal_neg_pi_div_two]
|
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 | 604 | 604 | theorem toReal_eq_pi_iff {θ : Angle} : θ.toReal = π ↔ θ = π := by | rw [← toReal_inj, toReal_pi]
|
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
variable {T : Type u} [Category.{v} T]
... | Mathlib/CategoryTheory/Comma/Arrow.lean | 86 | 88 | theorem mk_eq (f : Arrow T) : Arrow.mk f.hom = f := by |
cases f
rfl
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 265 | 265 | theorem relindex_bot_right : H.relindex ⊥ = 1 := by | rw [relindex, subgroupOf_bot_eq_top, index_top]
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 28 | 29 | theorem zipWith_distrib_tail : (zipWith f l l').tail = zipWith f l.tail l'.tail := by |
rw [← drop_one]; simp [zipWith_distrib_drop]
|
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 70 | 71 | theorem dist_eq_zero_iff_eq_or_not_reachable {u v : V} :
G.dist u v = 0 ↔ u = v ∨ ¬G.Reachable u v := by | simp [dist, Nat.sInf_eq_zero, Reachable]
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 262 | 262 | theorem degree_one_le : degree (1 : R[X]) ≤ (0 : WithBot ℕ) := by | rw [← C_1]; exact degree_C_le
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 545 | 546 | theorem map_objs_eq (hφ : Function.Injective φ.obj) : (map φ hφ S).objs = φ.obj '' S.objs := by |
ext x; convert mem_map_objs_iff S φ hφ x
|
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 793 | 830 | theorem uniformIntegrable_of' [IsFiniteMeasure μ] (hp : 1 ≤ p) (hp' : p ≠ ∞)
(hf : ∀ i, StronglyMeasurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0,
∀ i, snorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UniformIntegrable f p μ := by |
refine ⟨fun i => (hf i).aestronglyMeasurable,
unifIntegrable_of hp hp' (fun i => (hf i).aestronglyMeasurable) h, ?_⟩
obtain ⟨C, hC⟩ := h 1 one_pos
refine ⟨((C : ℝ≥0∞) * μ Set.univ ^ p.toReal⁻¹ + 1).toNNReal, fun i => ?_⟩
calc
snorm (f i) p μ ≤
snorm ({ x : α | ‖f i x‖₊ < C }.indicator (f i)) p ... |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 131 | 135 | theorem lengthParity_eq_ofAdd_length (w : W) :
cs.lengthParity w = Multiplicative.ofAdd (↑(ℓ w)) := by |
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rw [← hω, wordProd, map_list_prod, List.map_map, lengthParity_comp_simple, map_const',
prod_replicate, ← ofAdd_nsmul, nsmul_one]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 320 | 322 | theorem coprime_iff_isRelPrime {m n : ℕ} : m.Coprime n ↔ IsRelPrime m n := by |
simp_rw [coprime_iff_gcd_eq_one, IsRelPrime, ← and_imp, ← dvd_gcd_iff, isUnit_iff_dvd_one]
exact ⟨fun h _ ↦ (h ▸ ·), (dvd_one.mp <| · dvd_rfl)⟩
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 242 | 245 | theorem iSup_lintegral_le {ι : Sort*} (f : ι → α → ℝ≥0∞) :
⨆ i, ∫⁻ a, f i a ∂μ ≤ ∫⁻ a, ⨆ i, f i a ∂μ := by |
simp only [← iSup_apply]
exact (monotone_lintegral μ).le_map_iSup
|
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 299 | 303 | theorem compProd_null (a : α) (hs : MeasurableSet s) :
(κ ⊗ₖ η) a s = 0 ↔ (fun b => η (a, b) (Prod.mk b ⁻¹' s)) =ᵐ[κ a] 0 := by |
rw [kernel.compProd_apply _ _ _ hs, lintegral_eq_zero_iff]
· rfl
· exact kernel.measurable_kernel_prod_mk_left' hs a
|
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 85 | 89 | theorem sign_neg {r : ℝ} : sign (-r) = -sign r := by |
obtain hn | rfl | hp := lt_trichotomy r (0 : ℝ)
· rw [sign_of_neg hn, sign_of_pos (neg_pos.mpr hn), neg_neg]
· rw [sign_zero, neg_zero, sign_zero]
· rw [sign_of_pos hp, sign_of_neg (neg_lt_zero.mpr hp)]
|
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 | 51 | 54 | theorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1 := by |
by_cases h : a = 0
· simp only [opow_def, if_pos h, sub_zero]
· simp only [opow_def, if_neg h, limitRecOn_zero]
|
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