Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | rank int64 0 2.4k |
|---|---|---|---|---|---|---|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 162 | 166 | theorem DifferentiableAt.conformalAt (h : DifferentiableAt β f z) (hf' : deriv f z β 0) :
ConformalAt f z := by |
rw [conformalAt_iff_isConformalMap_fderiv, (h.hasFDerivAt.restrictScalars β).fderiv]
apply isConformalMap_complex_linear
simpa only [Ne, ext_ring_iff]
| 1,872 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 171 | 185 | theorem conformalAt_iff_differentiableAt_or_differentiableAt_comp_conj {f : β β β} {z : β} :
ConformalAt f z β
(DifferentiableAt β f z β¨ DifferentiableAt β (f β conj) (conj z)) β§ fderiv β f z β 0 := by |
rw [conformalAt_iff_isConformalMap_fderiv]
rw [isConformalMap_iff_is_complex_or_conj_linear]
apply and_congr_left
intro h
have h_diff := h.imp_symm fderiv_zero_of_not_differentiableAt
apply or_congr
Β· rw [differentiableAt_iff_restrictScalars β h_diff]
rw [β conj_conj z] at h_diff
rw [differentiableAt... | 1,872 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {π E F : Type*}
[NontriviallyNormedField π] [CompleteSpace π]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 42 | 102 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar' (h : Function.Surjective L) :
β (c : ββ₯0β), 0 < c β§ c < β β§ ΞΌ.map L = (c * addHaar (univ : Set (LinearMap.ker L))) β’ Ξ½ := by |
/- This is true for the second projection in product spaces, as the projection of the Haar
measure `ΞΌS.prod ΞΌT` is equal to the Haar measure `ΞΌT` multiplied by the total mass of `ΞΌS`. This
is also true for linear equivalences, as they map Haar measure to Haar measure. The general case
follows from these two an... | 1,873 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {π E F : Type*}
[NontriviallyNormedField π] [CompleteSpace π]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 106 | 109 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
β (c : ββ₯0β), 0 < c β§ ΞΌ.map L = c β’ Ξ½ := by |
rcases L.exists_map_addHaar_eq_smul_addHaar' ΞΌ Ξ½ h with β¨c, c_pos, -, hcβ©
exact β¨_, by simp [c_pos, NeZero.ne addHaar], hcβ©
| 1,873 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Geometry.Manifold.ChartedSpace
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.ContDiff.Basic
... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 256 | 258 | theorem target_eq : I.target = range (I : H β E) := by |
rw [β image_univ, β I.source_eq]
exact I.image_source_eq_target.symm
| 1,874 |
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 | 97 | 100 | theorem contDiffWithinAtProp_self_source {f : E β H'} {s : Set E} {x : E} :
ContDiffWithinAtProp π(π, E) I' n f s x β ContDiffWithinAt π n (I' β f) s x := by |
simp_rw [ContDiffWithinAtProp, modelWithCornersSelf_coe, range_id, inter_univ,
modelWithCornersSelf_coe_symm, CompTriple.comp_eq, preimage_id_eq, id_eq]
| 1,875 |
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 | 116 | 154 | theorem contDiffWithinAt_localInvariantProp (n : ββ) :
(contDiffGroupoid β I).LocalInvariantProp (contDiffGroupoid β I')
(ContDiffWithinAtProp I I' n) where
is_local {s x u f} u_open xu := by |
have : I.symm β»ΒΉ' (s β© u) β© range I = I.symm β»ΒΉ' s β© range I β© I.symm β»ΒΉ' u := by
simp only [inter_right_comm, preimage_inter]
rw [ContDiffWithinAtProp, ContDiffWithinAtProp, this]
symm
apply contDiffWithinAt_inter
have : u β π (I.symm (I x)) := by
rw [ModelWithCorners.left_inv]
... | 1,875 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 52 | 77 | theorem ContMDiffWithinAt.comp {t : Set M'} {g : M' β M''} (x : M)
(hg : ContMDiffWithinAt I' I'' n g t (f x)) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) : ContMDiffWithinAt I I'' n (g β f) s x := by |
rw [contMDiffWithinAt_iff] at hg hf β’
refine β¨hg.1.comp hf.1 st, ?_β©
set e := extChartAt I x
set e' := extChartAt I' (f x)
have : e' (f x) = (writtenInExtChartAt I I' x f) (e x) := by simp only [e, e', mfld_simps]
rw [this] at hg
have A : βαΆ y in π[e.symm β»ΒΉ' s β© range I] e x, f (e.symm y) β t β§ f (e.sy... | 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 81 | 84 | theorem ContMDiffWithinAt.comp_of_eq {t : Set M'} {g : M' β M''} {x : M} {y : M'}
(hg : ContMDiffWithinAt I' I'' n g t y) (hf : ContMDiffWithinAt I I' n f s x)
(st : MapsTo f s t) (hx : f x = y) : ContMDiffWithinAt I I'' n (g β f) s x := by |
subst hx; exact hg.comp x hf st
| 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 119 | 122 | theorem ContMDiff.comp {g : M' β M''} (hg : ContMDiff I' I'' n g) (hf : ContMDiff I I' n f) :
ContMDiff I I'' n (g β f) := by |
rw [β contMDiffOn_univ] at hf hg β’
exact hg.comp hf subset_preimage_univ
| 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 244 | 248 | theorem contMDiff_const : ContMDiff I I' n fun _ : M => c := by |
intro x
refine β¨continuousWithinAt_const, ?_β©
simp only [ContDiffWithinAtProp, (Β· β Β·)]
exact contDiffWithinAt_const
| 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 252 | 253 | theorem contMDiff_one [One M'] : ContMDiff I I' n (1 : M β M') := by |
simp only [Pi.one_def, contMDiff_const]
| 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M β M') := by | simp only [Pi.one_def, smooth_const]
| 1,876 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 59 | 63 | theorem ContMDiffWithinAt.prod_mk {f : M β M'} {g : M β N'} (hf : ContMDiffWithinAt I I' n f s x)
(hg : ContMDiffWithinAt I J' n g s x) :
ContMDiffWithinAt I (I'.prod J') n (fun x => (f x, g x)) s x := by |
rw [contMDiffWithinAt_iff] at *
exact β¨hf.1.prod hg.1, hf.2.prod hg.2β©
| 1,877 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 66 | 70 | theorem ContMDiffWithinAt.prod_mk_space {f : M β E'} {g : M β F'}
(hf : ContMDiffWithinAt I π(π, E') n f s x) (hg : ContMDiffWithinAt I π(π, F') n g s x) :
ContMDiffWithinAt I π(π, E' Γ F') n (fun x => (f x, g x)) s x := by |
rw [contMDiffWithinAt_iff] at *
exact β¨hf.1.prod hg.1, hf.2.prod hg.2β©
| 1,877 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 149 | 162 | theorem contMDiffWithinAt_fst {s : Set (M Γ N)} {p : M Γ N} :
ContMDiffWithinAt (I.prod J) I n Prod.fst s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' β¨continuousWithinAt_fst, _β©
refine' contDiffWithinAt_fst.congr (fun y hy => _) _
Β· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
Β· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... | 1,877 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 218 | 231 | theorem contMDiffWithinAt_snd {s : Set (M Γ N)} {p : M Γ N} :
ContMDiffWithinAt (I.prod J) J n Prod.snd s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' β¨continuousWithinAt_snd, _β©
refine' contDiffWithinAt_snd.congr (fun y hy => _) _
Β· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
Β· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... | 1,877 |
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 51 | 55 | theorem contMDiffWithinAt_iff_contDiffWithinAt {f : E β E'} {s : Set E} {x : E} :
ContMDiffWithinAt π(π, E) π(π, E') n f s x β ContDiffWithinAt π n f s x := by |
simp (config := { contextual := true }) only [ContMDiffWithinAt, liftPropWithinAt_iff',
ContDiffWithinAtProp, iff_def, mfld_simps]
exact ContDiffWithinAt.continuousWithinAt
| 1,878 |
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 63 | 65 | theorem contMDiffAt_iff_contDiffAt {f : E β E'} {x : E} :
ContMDiffAt π(π, E) π(π, E') n f x β ContDiffAt π n f x := by |
rw [β contMDiffWithinAt_univ, contMDiffWithinAt_iff_contDiffWithinAt, contDiffWithinAt_univ]
| 1,878 |
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 81 | 82 | theorem contMDiff_iff_contDiff {f : E β E'} : ContMDiff π(π, E) π(π, E') n f β ContDiff π n f := by |
rw [β contDiffOn_univ, β contMDiffOn_univ, contMDiffOn_iff_contDiffOn]
| 1,878 |
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.vector_bundle.fiberwise_linear from "leanprover-community/mathlib"@"be2c24f56783935652cefffb4bfca7e4b25d167e"
noncomputable section
open Set TopologicalSpace
open scoped Manifold Topology
variable {π B F : Type*} [Topolog... | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | 74 | 82 | theorem source_trans_partialHomeomorph (hU : IsOpen U)
(hΟ : ContinuousOn (fun x => Ο x : B β F βL[π] F) U)
(h2Ο : ContinuousOn (fun x => (Ο x).symm : B β F βL[π] F) U) (hU' : IsOpen U')
(hΟ' : ContinuousOn (fun x => Ο' x : B β F βL[π] F) U')
(h2Ο' : ContinuousOn (fun x => (Ο' x).symm : B β F βL[π] ... |
dsimp only [FiberwiseLinear.partialHomeomorph]; mfld_set_tac
| 1,879 |
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
#align_import geometry.manifold.vector_bundle.fiberwise_linear from "leanprover-community/mathlib"@"be2c24f56783935652cefffb4bfca7e4b25d167e"
noncomputable section
open Set TopologicalSpace
open scoped Manifold Topology
variable {π B F : Type*} [Topolog... | Mathlib/Geometry/Manifold/VectorBundle/FiberwiseLinear.lean | 87 | 95 | theorem target_trans_partialHomeomorph (hU : IsOpen U)
(hΟ : ContinuousOn (fun x => Ο x : B β F βL[π] F) U)
(h2Ο : ContinuousOn (fun x => (Ο x).symm : B β F βL[π] F) U) (hU' : IsOpen U')
(hΟ' : ContinuousOn (fun x => Ο' x : B β F βL[π] F) U')
(h2Ο' : ContinuousOn (fun x => (Ο' x).symm : B β F βL[π] ... |
dsimp only [FiberwiseLinear.partialHomeomorph]; mfld_set_tac
| 1,879 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 36 | 42 | theorem contMDiff_model : ContMDiff I π(π, E) n I := by |
intro x
refine (contMDiffAt_iff _ _).mpr β¨I.continuousAt, ?_β©
simp only [mfld_simps]
refine contDiffWithinAt_id.congr_of_eventuallyEq ?_ ?_
Β· exact Filter.eventuallyEq_of_mem self_mem_nhdsWithin fun xβ => I.right_inv
simp_rw [Function.comp_apply, I.left_inv, Function.id_def]
| 1,880 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 45 | 49 | theorem contMDiffOn_model_symm : ContMDiffOn π(π, E) I n I.symm (range I) := by |
rw [contMDiffOn_iff]
refine β¨I.continuousOn_symm, fun x y => ?_β©
simp only [mfld_simps]
exact contDiffOn_id.congr fun x' => I.right_inv
| 1,880 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 105 | 110 | theorem contMDiffOn_extend_symm (he : e β maximalAtlas I M) :
ContMDiffOn π(π, E) I n (e.extend I).symm (I '' e.target) := by |
refine (contMDiffOn_symm_of_mem_maximalAtlas he).comp
(contMDiffOn_model_symm.mono <| image_subset_range _ _) ?_
simp_rw [image_subset_iff, PartialEquiv.restr_coe_symm, I.toPartialEquiv_coe_symm,
preimage_preimage, I.left_inv, preimage_id']; rfl
| 1,880 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {π : Type*} [NontriviallyNormedField π]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace π E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 113 | 116 | theorem contMDiffOn_extChartAt_symm (x : M) :
ContMDiffOn π(π, E) I n (extChartAt I x).symm (extChartAt I x).target := by |
convert contMDiffOn_extend_symm (chart_mem_maximalAtlas I x)
rw [extChartAt_target, I.image_eq]
| 1,880 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 108 | 114 | theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
chartAt (ModelProd HB F) x =
(trivializationAt F E x.proj).toPartialHomeomorph β«β
(chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by |
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
chartAt_self_eq]
rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]
| 1,881 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 117 | 121 | theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F)
(hy : y β (chartAt (ModelProd HB F) x).target) :
((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1 := by |
simp only [FiberBundle.chartedSpace_chartAt, mfld_simps] at hy β’
exact (trivializationAt F E x.proj).proj_symm_apply hy.2
| 1,881 |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 178 | 196 | theorem contMDiffWithinAt_totalSpace (f : M β TotalSpace F E) {s : Set M} {xβ : M} :
ContMDiffWithinAt IM (IB.prod π(π, F)) n f s xβ β
ContMDiffWithinAt IM IB n (fun x => (f x).proj) s xβ β§
ContMDiffWithinAt IM π(π, F) n (fun x β¦ (trivializationAt F E (f xβ).proj (f x)).2) s xβ := by |
simp (config := { singlePass := true }) only [contMDiffWithinAt_iff_target]
rw [and_and_and_comm, β FiberBundle.continuousWithinAt_totalSpace, and_congr_right_iff]
intro hf
simp_rw [modelWithCornersSelf_prod, FiberBundle.extChartAt, Function.comp,
PartialEquiv.trans_apply, PartialEquiv.prod_coe, PartialEqu... | 1,881 |
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.Metrizable.Urysohn
#align_import geometry.manifold.metrizable from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
open TopologicalSpace
| Mathlib/Geometry/Manifold/Metrizable.lean | 24 | 31 | theorem ManifoldWithCorners.metrizableSpace {E : Type*} [NormedAddCommGroup E] [NormedSpace β E]
[FiniteDimensional β E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners β E H)
(M : Type*) [TopologicalSpace M] [ChartedSpace H M] [SigmaCompactSpace M] [T2Space M] :
MetrizableSpace M := by |
haveI := I.locallyCompactSpace; haveI := ChartedSpace.locallyCompactSpace H M
haveI := I.secondCountableTopology
haveI := ChartedSpace.secondCountable_of_sigma_compact H M
exact metrizableSpace_of_t3_second_countable M
| 1,882 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 133 | 141 | theorem isOpen_A (L : E βL[π] F) (r Ξ΅ : β) : IsOpen (A f L r Ξ΅) := by |
rw [Metric.isOpen_iff]
rintro x β¨r', r'_mem, hr'β©
obtain β¨s, s_gt, s_ltβ© : β s : β, r / 2 < s β§ s < r' := exists_between r'_mem.1
have : s β Ioc (r / 2) r := β¨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)β©
refine β¨r' - s, by linarith, fun x' hx' => β¨s, this, ?_β©β©
have B : ball x' s β ball x r' := ball_subset (le... | 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 144 | 145 | theorem isOpen_B {K : Set (E βL[π] F)} {r s Ξ΅ : β} : IsOpen (B f K r s Ξ΅) := by |
simp [B, isOpen_biUnion, IsOpen.inter, isOpen_A]
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 148 | 151 | theorem A_mono (L : E βL[π] F) (r : β) {Ξ΅ Ξ΄ : β} (h : Ξ΅ β€ Ξ΄) : A f L r Ξ΅ β A f L r Ξ΄ := by |
rintro x β¨r', r'r, hr'β©
refine β¨r', r'r, fun y hy z hz => (hr' y hy z hz).trans_le (mul_le_mul_of_nonneg_right h ?_)β©
linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x]
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 154 | 159 | theorem le_of_mem_A {r Ξ΅ : β} {L : E βL[π] F} {x : E} (hx : x β A f L r Ξ΅) {y z : E}
(hy : y β closedBall x (r / 2)) (hz : z β closedBall x (r / 2)) :
βf z - f y - L (z - y)β β€ Ξ΅ * r := by |
rcases hx with β¨r', r'mem, hr'β©
apply le_of_lt
exact hr' _ ((mem_closedBall.1 hy).trans_lt r'mem.1) _ ((mem_closedBall.1 hz).trans_lt r'mem.1)
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 162 | 181 | theorem mem_A_of_differentiable {Ξ΅ : β} (hΞ΅ : 0 < Ξ΅) {x : E} (hx : DifferentiableAt π f x) :
β R > 0, β r β Ioo (0 : β) R, x β A f (fderiv π f x) r Ξ΅ := by |
let Ξ΄ := (Ξ΅ / 2) / 2
obtain β¨R, R_pos, hRβ© :
β R > 0, β y β ball x R, βf y - f x - fderiv π f x (y - x)β β€ Ξ΄ * βy - xβ :=
eventually_nhds_iff_ball.1 <| hx.hasFDerivAt.isLittleO.bound <| by positivity
refine β¨R, R_pos, fun r hr => ?_β©
have : r β Ioc (r / 2) r := right_mem_Ioc.2 <| half_lt_self hr.1
... | 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 473 | 483 | theorem A_mem_nhdsWithin_Ioi {L : F} {r Ξ΅ x : β} (hx : x β A f L r Ξ΅) : A f L r Ξ΅ β π[>] x := by |
rcases hx with β¨r', rr', hr'β©
rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset]
obtain β¨s, s_gt, s_ltβ© : β s : β, r / 2 < s β§ s < r' := exists_between rr'.1
have : s β Ioc (r / 2) r := β¨s_gt, le_of_lt (s_lt.trans_le rr'.2)β©
refine β¨x + r' - s, by simp only [mem_Ioi]; linarith, fun x' hx' => β¨s, this, ?_β©β©
have... | 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 486 | 492 | theorem B_mem_nhdsWithin_Ioi {K : Set F} {r s Ξ΅ x : β} (hx : x β B f K r s Ξ΅) :
B f K r s Ξ΅ β π[>] x := by |
obtain β¨L, LK, hLβ, hLββ© : β L : F, L β K β§ x β A f L r Ξ΅ β§ x β A f L s Ξ΅ := by
simpa only [B, mem_iUnion, mem_inter_iff, exists_prop] using hx
filter_upwards [A_mem_nhdsWithin_Ioi hLβ, A_mem_nhdsWithin_Ioi hLβ] with y hyβ hyβ
simp only [B, mem_iUnion, mem_inter_iff, exists_prop]
exact β¨L, LK, hyβ, hyββ©
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 499 | 502 | theorem A_mono (L : F) (r : β) {Ξ΅ Ξ΄ : β} (h : Ξ΅ β€ Ξ΄) : A f L r Ξ΅ β A f L r Ξ΄ := by |
rintro x β¨r', r'r, hr'β©
refine β¨r', r'r, fun y hy z hz => (hr' y hy z hz).trans (mul_le_mul_of_nonneg_right h ?_)β©
linarith [hy.1, hy.2, r'r.2]
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 505 | 510 | theorem le_of_mem_A {r Ξ΅ : β} {L : F} {x : β} (hx : x β A f L r Ξ΅) {y z : β}
(hy : y β Icc x (x + r / 2)) (hz : z β Icc x (x + r / 2)) :
βf z - f y - (z - y) β’ Lβ β€ Ξ΅ * r := by |
rcases hx with β¨r', r'mem, hr'β©
have A : x + r / 2 β€ x + r' := by linarith [r'mem.1]
exact hr' _ ((Icc_subset_Icc le_rfl A) hy) _ ((Icc_subset_Icc le_rfl A) hz)
| 1,883 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 513 | 538 | theorem mem_A_of_differentiable {Ξ΅ : β} (hΞ΅ : 0 < Ξ΅) {x : β}
(hx : DifferentiableWithinAt β f (Ici x) x) :
β R > 0, β r β Ioo (0 : β) R, x β A f (derivWithin f (Ici x) x) r Ξ΅ := by |
have := hx.hasDerivWithinAt
simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
rcases mem_nhdsWithin_Ici_iff_exists_Ico_subset.1 (this (half_pos hΞ΅)) with β¨m, xm, hmβ©
refine β¨m - x, by linarith [show x < m from xm], fun r hr => ?_β©
have : r β Ioc (r / 2) r := β¨half_lt_self hr.1, le_rflβ©
refine... | 1,883 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 33 | 38 | theorem measurableSet_lineDifferentiableAt (hf : Continuous f) :
MeasurableSet {x : E | LineDifferentiableAt π f x v} := by |
borelize π
let g : E β π β F := fun x t β¦ f (x + t β’ v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact measurable_prod_mk_right (measurableSet_of_differentiableAt_with_param π hg)
| 1,884 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 40 | 45 | theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun x β¦ lineDeriv π f x v) := by |
borelize π
let g : E β π β F := fun x t β¦ f (x + t β’ v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (measurable_deriv_with_param hg).comp measurable_prod_mk_right
| 1,884 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 47 | 52 | theorem stronglyMeasurable_lineDeriv [SecondCountableTopologyEither E F] (hf : Continuous f) :
StronglyMeasurable (fun x β¦ lineDeriv π f x v) := by |
borelize π
let g : E β π β F := fun x t β¦ f (x + t β’ v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (stronglyMeasurable_deriv_with_param hg).comp_measurable measurable_prod_mk_right
| 1,884 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 72 | 81 | theorem measurableSet_lineDifferentiableAt_uncurry (hf : Continuous f) :
MeasurableSet {p : E Γ E | LineDifferentiableAt π f p.1 p.2} := by |
borelize π
let g : (E Γ E) β π β F := fun p t β¦ f (p.1 + t β’ p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
have M_meas : MeasurableSet {q : (E Γ E) Γ π | DifferentiableAt π (g q.1) q.2} :=
meas... | 1,884 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 83 | 90 | theorem measurable_lineDeriv_uncurry [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun (p : E Γ E) β¦ lineDeriv π f p.1 p.2) := by |
borelize π
let g : (E Γ E) β π β F := fun p t β¦ f (p.1 + t β’ p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (measurable_deriv_with_param this).comp measurable_prod_mk_right
| 1,884 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {π : Type*} [NontriviallyNormedField π] [LocallyCompactSpace π]
{E : Type*} [NormedAddCommGroup E] [NormedSpace π E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 92 | 99 | theorem stronglyMeasurable_lineDeriv_uncurry (hf : Continuous f) :
StronglyMeasurable (fun (p : E Γ E) β¦ lineDeriv π f p.1 p.2) := by |
borelize π
let g : (E Γ E) β π β F := fun p t β¦ f (p.1 + t β’ p.2)
have : Continuous g.uncurry :=
hf.comp <| (continuous_fst.comp continuous_fst).add
<| continuous_snd.smul (continuous_snd.comp continuous_fst)
exact (stronglyMeasurable_deriv_with_param this).comp_measurable measurable_prod_mk_right
| 1,884 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 118 | 120 | theorem one_lt_rOut_div_rIn {c : E} (f : ContDiffBump c) : 1 < f.rOut / f.rIn := by |
rw [one_lt_div f.rIn_pos]
exact f.rIn_lt_rOut
| 1,885 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 154 | 157 | theorem one_of_mem_closedBall (hx : x β closedBall c f.rIn) : f x = 1 := by |
apply ContDiffBumpBase.eq_one _ _ f.one_lt_rOut_div_rIn
simpa only [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_nonneg f.rIn_pos.le, β div_eq_inv_mul,
div_le_one f.rIn_pos] using mem_closedBall_iff_norm.1 hx
| 1,885 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 172 | 176 | theorem support_eq : Function.support f = Metric.ball c f.rOut := by |
simp only [toFun, support_comp_eq_preimage, ContDiffBumpBase.support _ _ f.one_lt_rOut_div_rIn]
ext x
simp only [mem_ball_iff_norm, sub_zero, norm_smul, mem_preimage, Real.norm_eq_abs, abs_inv,
abs_of_pos f.rIn_pos, β div_eq_inv_mul, div_lt_div_right f.rIn_pos]
| 1,885 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Function Set Filter
open scoped Topology Filter
variable... | Mathlib/Analysis/Calculus/BumpFunction/Basic.lean | 179 | 180 | theorem tsupport_eq : tsupport f = closedBall c f.rOut := by |
simp_rw [tsupport, f.support_eq, closure_ball _ f.rOut_pos.ne']
| 1,885 |
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.RCLike.Basic
#align_import data.is_R_or_C.lemmas from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
variable {K E : Type*} [RCLike K]
namespace RCLike
@[simp, rclike_simps]
| Mathlib/Analysis/RCLike/Lemmas.lean | 71 | 74 | theorem reCLM_norm : β(reCLM : K βL[β] β)β = 1 := by |
apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
convert ContinuousLinearMap.ratio_le_opNorm (reCLM : K βL[β] β) (1 : K)
simp
| 1,886 |
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 44 | 59 | theorem exists_extension_norm_eq (p : Subspace β E) (f : p βL[β] β) :
β g : E βL[β] β, (β x : p, g x = f x) β§ βgβ = βfβ := by |
rcases exists_extension_of_le_sublinear β¨p, fβ© (fun x => βfβ * βxβ)
(fun c hc x => by simp only [norm_smul c x, Real.norm_eq_abs, abs_of_pos hc, mul_left_comm])
(fun x y => by -- Porting note: placeholder filled here
rw [β left_distrib]
exact mul_le_mul_of_nonneg_left (norm_add_le x y) (@... | 1,887 |
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.Analysis.RCLike.Lemmas
#align_import analysis.normed_space.hahn_banach.extension from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
univers... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 151 | 163 | theorem coord_norm' {x : E} (h : x β 0) : β(βxβ : π) β’ coord π x hβ = 1 := by |
#adaptation_note
/--
`set_option maxSynthPendingDepth 2` required after https://github.com/leanprover/lean4/pull/4119
Alternatively, we can add:
```
let X : SeminormedAddCommGroup (β₯(span π {x}) βL[π] π) := inferInstance
have : BoundedSMul π (β₯(span π {x}) βL[π] π) := @NormedSpace.boundedSMul π _... | 1,887 |
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 82 | 84 | theorem inclusionInDoubleDual_norm_le : βinclusionInDoubleDual π Eβ β€ 1 := by |
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
| 1,888 |
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 87 | 88 | theorem double_dual_bound (x : E) : β(inclusionInDoubleDual π E) xβ β€ βxβ := by |
simpa using ContinuousLinearMap.le_of_opNorm_le _ (inclusionInDoubleDual_norm_le π E) x
| 1,888 |
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 101 | 103 | theorem dualPairing_separatingLeft : (dualPairing π E).SeparatingLeft := by |
rw [LinearMap.separatingLeft_iff_ker_eq_bot, LinearMap.ker_eq_bot]
exact ContinuousLinearMap.coe_injective
| 1,888 |
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
#align_import analysis.normed_space.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Filter Function Bornology Metric Set
open Topology Filter... | Mathlib/Analysis/NormedSpace/WeakDual.lean | 141 | 145 | theorem dual_norm_topology_le_weak_dual_topology :
(UniformSpace.toTopologicalSpace : TopologicalSpace (Dual π E)) β€
(WeakDual.instTopologicalSpace : TopologicalSpace (WeakDual π E)) := by |
convert (@toWeakDual_continuous _ _ _ _ (by assumption)).le_induced
exact induced_id.symm
| 1,889 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.is_R_or_C from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
noncomputable section
open NNReal ENNReal
namespace RCLike
variabl... | Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean | 73 | 77 | theorem measurable_of_re_im (hre : Measurable fun x => RCLike.re (f x))
(him : Measurable fun x => RCLike.im (f x)) : Measurable f := by |
convert Measurable.add (M := π) (RCLike.measurable_ofReal.comp hre)
((RCLike.measurable_ofReal.comp him).mul_const RCLike.I)
exact (RCLike.re_add_im _).symm
| 1,890 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Complex
#align_import measure_theory.function.special_functions.is_R_or_C from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
noncomputable section
open NNReal ENNReal
namespace RCLike
variabl... | Mathlib/MeasureTheory/Function/SpecialFunctions/RCLike.lean | 80 | 84 | theorem aemeasurable_of_re_im (hre : AEMeasurable (fun x => RCLike.re (f x)) ΞΌ)
(him : AEMeasurable (fun x => RCLike.im (f x)) ΞΌ) : AEMeasurable f ΞΌ := by |
convert AEMeasurable.add (M := π) (RCLike.measurable_ofReal.comp_aemeasurable hre)
((RCLike.measurable_ofReal.comp_aemeasurable him).mul_const RCLike.I)
exact (RCLike.re_add_im _).symm
| 1,890 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 42 | 43 | theorem Memβp.integrable_sq {f : Ξ± β β} (h : Memβp f 2 ΞΌ) : Integrable (fun x => f x ^ 2) ΞΌ := by |
simpa [β memβp_one_iff_integrable] using h.norm_rpow two_ne_zero ENNReal.two_ne_top
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 46 | 51 | theorem memβp_two_iff_integrable_sq_norm {f : Ξ± β F} (hf : AEStronglyMeasurable f ΞΌ) :
Memβp f 2 ΞΌ β Integrable (fun x => βf xβ ^ 2) ΞΌ := by |
rw [β memβp_one_iff_integrable]
convert (memβp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm
Β· simp
Β· rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top]
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 54 | 57 | theorem memβp_two_iff_integrable_sq {f : Ξ± β β} (hf : AEStronglyMeasurable f ΞΌ) :
Memβp f 2 ΞΌ β Integrable (fun x => f x ^ 2) ΞΌ := by |
convert memβp_two_iff_integrable_sq_norm hf using 3
simp
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 81 | 83 | theorem Integrable.const_inner (c : E) (hf : Integrable f ΞΌ) :
Integrable (fun x => βͺc, f xβ«) ΞΌ := by |
rw [β memβp_one_iff_integrable] at hf β’; exact hf.const_inner c
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 86 | 88 | theorem Integrable.inner_const (hf : Integrable f ΞΌ) (c : E) :
Integrable (fun x => βͺf x, cβ«) ΞΌ := by |
rw [β memβp_one_iff_integrable] at hf β’; exact hf.inner_const c
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 104 | 106 | theorem _root_.integral_eq_zero_of_forall_integral_inner_eq_zero (f : Ξ± β E) (hf : Integrable f ΞΌ)
(hf_int : β c : E, β« x, βͺc, f xβ« βΞΌ = 0) : β« x, f x βΞΌ = 0 := by |
specialize hf_int (β« x, f x βΞΌ); rwa [integral_inner hf, inner_self_eq_zero] at hf_int
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 118 | 121 | theorem snorm_rpow_two_norm_lt_top (f : Lp F 2 ΞΌ) : snorm (fun x => βf xβ ^ (2 : β)) 1 ΞΌ < β := by |
have h_two : ENNReal.ofReal (2 : β) = 2 := by simp [zero_le_one]
rw [snorm_norm_rpow f zero_lt_two, one_mul, h_two]
exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.snorm_ne_top f)
| 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 124 | 140 | theorem snorm_inner_lt_top (f g : Ξ± ββ[ΞΌ] E) : snorm (fun x : Ξ± => βͺf x, g xβ«) 1 ΞΌ < β := by |
have h : β x, ββͺf x, g xβ«β β€ ββf xβ ^ (2 : β) + βg xβ ^ (2 : β)β := by
intro x
rw [β @Nat.cast_two β, Real.rpow_natCast, Real.rpow_natCast]
calc
ββͺf x, g xβ«β β€ βf xβ * βg xβ := norm_inner_le_norm _ _
_ β€ 2 * βf xβ * βg xβ :=
(mul_le_mul_of_nonneg_right (le_mul_of_one_le_left (norm_non... | 1,891 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 154 | 167 | theorem integral_inner_eq_sq_snorm (f : Ξ± ββ[ΞΌ] E) :
β« a, βͺf a, f aβ« βΞΌ = ENNReal.toReal (β«β» a, (βf aββ : ββ₯0β) ^ (2 : β) βΞΌ) := by |
simp_rw [inner_self_eq_norm_sq_to_K]
norm_cast
rw [integral_eq_lintegral_of_nonneg_ae]
rotate_left
Β· exact Filter.eventually_of_forall fun x => sq_nonneg _
Β· exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable
congr
ext1 x
have h_two : (2 : β) = ((2 : β) : β) := by ... | 1,891 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 65 | 72 | theorem _root_.MeasureTheory.Memβp.evariance_lt_top [IsFiniteMeasure ΞΌ] (hX : Memβp X 2 ΞΌ) :
evariance X ΞΌ < β := by |
have := ENNReal.pow_lt_top (hX.sub <| memβp_const <| ΞΌ[X]).2 2
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top, β ENNReal.rpow_two] at this
simp only [coe_two, Pi.sub_apply, ENNReal.one_toReal, one_div] at this
rw [β ENNReal.rpow_mul, inv_mul_cancel (two_ne_zero : (2 : β) β 0), ENNReal.rpow_o... | 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 75 | 89 | theorem evariance_eq_top [IsFiniteMeasure ΞΌ] (hXm : AEStronglyMeasurable X ΞΌ) (hX : Β¬Memβp X 2 ΞΌ) :
evariance X ΞΌ = β := by |
by_contra h
rw [β Ne, β lt_top_iff_ne_top] at h
have : Memβp (fun Ο => X Ο - ΞΌ[X]) 2 ΞΌ := by
refine β¨hXm.sub aestronglyMeasurable_const, ?_β©
rw [snorm_eq_lintegral_rpow_nnnorm two_ne_zero ENNReal.two_ne_top]
simp only [coe_two, ENNReal.one_toReal, ENNReal.rpow_two, Ne]
exact ENNReal.rpow_lt_top_o... | 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 92 | 97 | theorem evariance_lt_top_iff_memβp [IsFiniteMeasure ΞΌ] (hX : AEStronglyMeasurable X ΞΌ) :
evariance X ΞΌ < β β Memβp X 2 ΞΌ := by |
refine β¨?_, MeasureTheory.Memβp.evariance_lt_topβ©
contrapose
rw [not_lt, top_le_iff]
exact evariance_eq_top hX
| 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 100 | 103 | theorem _root_.MeasureTheory.Memβp.ofReal_variance_eq [IsFiniteMeasure ΞΌ] (hX : Memβp X 2 ΞΌ) :
ENNReal.ofReal (variance X ΞΌ) = evariance X ΞΌ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
| 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 106 | 113 | theorem evariance_eq_lintegral_ofReal (X : Ξ© β β) (ΞΌ : Measure Ξ©) :
evariance X ΞΌ = β«β» Ο, ENNReal.ofReal ((X Ο - ΞΌ[X]) ^ 2) βΞΌ := by |
rw [evariance]
congr
ext1 Ο
rw [pow_two, β ENNReal.coe_mul, β nnnorm_mul, β pow_two]
congr
exact (Real.toNNReal_eq_nnnorm_of_nonneg <| sq_nonneg _).symm
| 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 116 | 125 | theorem _root_.MeasureTheory.Memβp.variance_eq_of_integral_eq_zero (hX : Memβp X 2 ΞΌ)
(hXint : ΞΌ[X] = 0) : variance X ΞΌ = ΞΌ[X ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, β ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)] <;>
simp_rw [hXint, sub_zero]
Β· rfl
Β· convert hX.integrable_norm_rpow two_ne_zero ENNReal.two_ne_top with Ο
simp only [Pi.sub_apply, Real.norm_eq_abs, coe_two, ENNReal.one_toRe... | 1,892 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 128 | 139 | theorem _root_.MeasureTheory.Memβp.variance_eq [IsFiniteMeasure ΞΌ] (hX : Memβp X 2 ΞΌ) :
variance X ΞΌ = ΞΌ[(X - fun _ => ΞΌ[X] :) ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, β ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)]
Β· rfl
Β· -- Porting note: `ΞΌ[X]` without whitespace is ambiguous as it could be GetElem,
-- and `convert` cannot disambiguate based on typeclass inference failure.
convert (hX.sub... | 1,892 |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 132 | 135 | theorem measure_mem_eq (h : IdentDistrib f g ΞΌ Ξ½) {s : Set Ξ³} (hs : MeasurableSet s) :
ΞΌ (f β»ΒΉ' s) = Ξ½ (g β»ΒΉ' s) := by |
rw [β Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, β
Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]
| 1,893 |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 141 | 145 | theorem ae_snd (h : IdentDistrib f g ΞΌ Ξ½) {p : Ξ³ β Prop} (pmeas : MeasurableSet {x | p x})
(hp : βα΅ x βΞΌ, p (f x)) : βα΅ x βΞ½, p (g x) := by |
apply (ae_map_iff h.aemeasurable_snd pmeas).1
rw [β h.map_eq]
exact (ae_map_iff h.aemeasurable_fst pmeas).2 hp
| 1,893 |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 162 | 168 | theorem aestronglyMeasurable_snd [TopologicalSpace Ξ³] [MetrizableSpace Ξ³] [BorelSpace Ξ³]
(h : IdentDistrib f g ΞΌ Ξ½) (hf : AEStronglyMeasurable f ΞΌ) : AEStronglyMeasurable g Ξ½ := by |
refine aestronglyMeasurable_iff_aemeasurable_separable.2 β¨h.aemeasurable_snd, ?_β©
rcases (aestronglyMeasurable_iff_aemeasurable_separable.1 hf).2 with β¨t, t_sep, htβ©
refine β¨closure t, t_sep.closure, ?_β©
apply h.ae_mem_snd isClosed_closure.measurableSet
filter_upwards [ht] with x hx using subset_closure hx
| 1,893 |
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open MeasureTheory Filter Finset
noncomputable section
open scoped Topology MeasureTheory ENNReal NNR... | Mathlib/Probability/IdentDistrib.lean | 326 | 348 | theorem Memβp.uniformIntegrable_of_identDistrib_aux {ΞΉ : Type*} {f : ΞΉ β Ξ± β E} {j : ΞΉ} {p : ββ₯0β}
(hp : 1 β€ p) (hp' : p β β) (hβp : Memβp (f j) p ΞΌ) (hfmeas : β i, StronglyMeasurable (f i))
(hf : β i, IdentDistrib (f i) (f j) ΞΌ ΞΌ) : UniformIntegrable f p ΞΌ := by |
refine uniformIntegrable_of' hp hp' hfmeas fun Ξ΅ hΞ΅ => ?_
by_cases hΞΉ : Nonempty ΞΉ
swap; Β· exact β¨0, fun i => False.elim (hΞΉ <| Nonempty.intro i)β©
obtain β¨C, hCβ, hCββ© := hβp.snorm_indicator_norm_ge_pos_le (hfmeas _) hΞ΅
refine β¨β¨C, hCβ.leβ©, fun i => le_trans (le_of_eq ?_) hCββ©
have : {x | (β¨C, hCβ.leβ© : ββ₯... | 1,893 |
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 | 47 | 76 | theorem separate_convex_open_set [TopologicalSpace E] [AddCommGroup E] [TopologicalAddGroup E]
[Module β E] [ContinuousSMul β E] {s : Set E} (hsβ : (0 : E) β s) (hsβ : Convex β s)
(hsβ : IsOpen s) {xβ : E} (hxβ : xβ β s) : β f : E βL[β] β, f xβ = 1 β§ β x β s, f x < 1 := by |
let f : E ββ.[β] β := LinearPMap.mkSpanSingleton xβ 1 (ne_of_mem_of_not_mem hsβ hxβ).symm
have := exists_extension_of_le_sublinear f (gauge s) (fun c hc => gauge_smul_of_nonneg hc.le)
(gauge_add_le hsβ <| absorbent_nhds_zero <| hsβ.mem_nhds hsβ) ?_
Β· obtain β¨Ο, hΟβ, hΟββ© := this
have hΟβ : Ο xβ = 1 := by... | 1,894 |
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 | 84 | 112 | theorem geometric_hahn_banach_open (hsβ : Convex β s) (hsβ : IsOpen s) (ht : Convex β t)
(disj : Disjoint s t) : β (f : E βL[β] β) (u : β), (β a β s, f a < u) β§ β b β t, u β€ f b := by |
obtain rfl | β¨aβ, haββ© := s.eq_empty_or_nonempty
Β· exact β¨0, 0, by simp, fun b _hb => le_rflβ©
obtain rfl | β¨bβ, hbββ© := t.eq_empty_or_nonempty
Β· exact β¨0, 1, fun a _ha => zero_lt_one, by simpβ©
let xβ := bβ - aβ
let C := xβ +α΅₯ (s - t)
have : (0 : E) β C :=
β¨aβ - bβ, sub_mem_sub haβ hbβ, by simp_rw [xβ... | 1,894 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 85 | 89 | theorem bottom_row_coprime {R : Type*} [CommRing R] (g : SL(2, R)) :
IsCoprime ((βg : Matrix (Fin 2) (Fin 2) R) 1 0) ((βg : Matrix (Fin 2) (Fin 2) R) 1 1) := by |
use -(βg : Matrix (Fin 2) (Fin 2) R) 0 1, (βg : Matrix (Fin 2) (Fin 2) R) 0 0
rw [add_comm, neg_mul, β sub_eq_add_neg, β det_fin_two]
exact g.det_coe
| 1,895 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 94 | 104 | theorem bottom_row_surj {R : Type*} [CommRing R] :
Set.SurjOn (fun g : SL(2, R) => (βg : Matrix (Fin 2) (Fin 2) R) 1) Set.univ
{cd | IsCoprime (cd 0) (cd 1)} := by |
rintro cd β¨bβ, a, gcd_eqnβ©
let A := of ![![a, -bβ], cd]
have det_A_1 : det A = 1 := by
convert gcd_eqn
rw [det_fin_two]
simp [A, (by ring : a * cd 1 + bβ * cd 0 = bβ * cd 0 + a * cd 1)]
refine β¨β¨A, det_A_1β©, Set.mem_univ _, ?_β©
ext; simp [A]
| 1,895 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 117 | 161 | theorem tendsto_normSq_coprime_pair :
Filter.Tendsto (fun p : Fin 2 β β€ => normSq ((p 0 : β) * z + p 1)) cofinite atTop := by |
-- using this instance rather than the automatic `Function.module` makes unification issues in
-- `LinearEquiv.closedEmbedding_of_injective` less bad later in the proof.
letI : Module β (Fin 2 β β) := NormedSpace.toModule
let Οβ : (Fin 2 β β) ββ[β] β := LinearMap.proj 0
let Οβ : (Fin 2 β β) ββ[β] β := Linear... | 1,895 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 90 | 97 | theorem coeff_invOfUnit [DecidableEq Ο] (n : Ο ββ β) (Ο : MvPowerSeries Ο R) (u : RΛ£) :
coeff R n (invOfUnit Ο u) =
if n = 0 then βuβ»ΒΉ
else
-βuβ»ΒΉ *
β x β antidiagonal n,
if x.2 < n then coeff R x.1 Ο * coeff R x.2 (invOfUnit Ο u) else 0 := by |
convert coeff_inv_aux n (βuβ»ΒΉ) Ο
| 1,896 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 101 | 104 | theorem constantCoeff_invOfUnit (Ο : MvPowerSeries Ο R) (u : RΛ£) :
constantCoeff Ο R (invOfUnit Ο u) = βuβ»ΒΉ := by |
classical
rw [β coeff_zero_eq_constantCoeff_apply, coeff_invOfUnit, if_pos rfl]
| 1,896 |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Fi... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 107 | 137 | theorem mul_invOfUnit (Ο : MvPowerSeries Ο R) (u : RΛ£) (h : constantCoeff Ο R Ο = u) :
Ο * invOfUnit Ο u = 1 :=
ext fun n =>
letI := Classical.decEq (Ο ββ β)
if H : n = 0 then by
rw [H]
simp [coeff_mul, support_single_ne_zero, h]
else by
classical
have : ((0 : Ο ββ β), n) β ant... | rw [mem_antidiagonal, zero_add]
rw [coeff_one, if_neg H, coeff_mul, β Finset.insert_erase this,
Finset.sum_insert (Finset.not_mem_erase _ _), coeff_zero_eq_constantCoeff_apply, h,
coeff_invOfUnit, if_neg H, neg_mul, mul_neg, Units.mul_inv_cancel_left, β
Finset.insert_erase this, Finset.su... | 1,896 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 36 | 57 | theorem exists_eq_polynomial [Semiring Fq] {d : β} {m : β} (hm : Fintype.card Fq ^ d β€ m)
(b : Fq[X]) (hb : natDegree b β€ d) (A : Fin m.succ β Fq[X])
(hA : β i, degree (A i) < degree b) : β iβ iβ, iβ β iβ β§ A iβ = A iβ := by |
-- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,
-- there must be two elements of A with the same coefficients at
-- `0`, ... `degree b - 1` β€ `d - 1`.
-- In other words, the following map is not injective:
set f : Fin m.succ β Fin d β Fq := fun i j => (A i).coef... | 1,897 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 63 | 98 | theorem exists_approx_polynomial_aux [Ring Fq] {d : β} {m : β} (hm : Fintype.card Fq ^ d β€ m)
(b : Fq[X]) (A : Fin m.succ β Fq[X]) (hA : β i, degree (A i) < degree b) :
β iβ iβ, iβ β iβ β§ degree (A iβ - A iβ) < β(natDegree b - d) := by |
have hb : b β 0 := by
rintro rfl
specialize hA 0
rw [degree_zero] at hA
exact not_lt_of_le bot_le hA
-- Since there are > q^d elements of A, and only q^d choices for the highest `d` coefficients,
-- there must be two elements of A with the same coefficients at
-- `degree b - 1`, ... `degree b -... | 1,897 |
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
import Mathlib.RingTheory.Ideal.LocalRing
#align_import number_theory.class_number.admissible_card_pow_degree from "leanprover-community/mathlib"@"0b... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 106 | 149 | theorem exists_approx_polynomial {b : Fq[X]} (hb : b β 0) {Ξ΅ : β} (hΞ΅ : 0 < Ξ΅)
(A : Fin (Fintype.card Fq ^ β-log Ξ΅ / log (Fintype.card Fq)ββ).succ β Fq[X]) :
β iβ iβ, iβ β iβ β§ (cardPowDegree (A iβ % b - A iβ % b) : β) < cardPowDegree b β’ Ξ΅ := by |
have hbΞ΅ : 0 < cardPowDegree b β’ Ξ΅ := by
rw [Algebra.smul_def, eq_intCast]
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hΞ΅
have one_lt_q : 1 < Fintype.card Fq := Fintype.one_lt_card
have one_lt_q' : (1 : β) < Fintype.card Fq := by assumption_mod_cast
have q_pos : 0 < Fintype.card Fq := by ... | 1,897 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import algebra.char_p.local_ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
| Mathlib/Algebra/CharP/LocalRing.lean | 25 | 67 | theorem charP_zero_or_prime_power (R : Type*) [CommRing R] [LocalRing R] (q : β)
[char_R_q : CharP R q] : q = 0 β¨ IsPrimePow q := by |
-- Assume `q := char(R)` is not zero.
apply or_iff_not_imp_left.2
intro q_pos
let K := LocalRing.ResidueField R
haveI RM_char := ringChar.charP K
let r := ringChar K
let n := q.factorization r
-- `r := char(R/m)` is either prime or zero:
cases' CharP.char_is_prime_or_zero K r with r_prime r_zero
Β· ... | 1,898 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 85 | 105 | theorem reduce_to_p_prime {P : Prop} :
(β p > 0, MixedCharZero R p β P) β β p : β, p.Prime β MixedCharZero R p β P := by |
constructor
Β· intro h q q_prime q_mixedChar
exact h q (Nat.Prime.pos q_prime) q_mixedChar
Β· intro h q q_pos q_mixedChar
rcases q_mixedChar.charP_quotient with β¨I, hI_ne_top, _β©
-- Krull's Thm: There exists a prime ideal `P` such that `I β€ P`
rcases Ideal.exists_le_maximal I hI_ne_top with β¨M, hM_... | 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 112 | 136 | theorem reduce_to_maximal_ideal {p : β} (hp : Nat.Prime p) :
(β I : Ideal R, I β β€ β§ CharP (R β§Έ I) p) β β I : Ideal R, I.IsMaximal β§ CharP (R β§Έ I) p := by |
constructor
Β· intro g
rcases g with β¨I, β¨hI_not_top, _β©β©
-- Krull's Thm: There exists a prime ideal `M` such that `I β€ M`.
rcases Ideal.exists_le_maximal I hI_not_top with β¨M, β¨hM_max, hM_geβ©β©
use M
constructor
Β· exact hM_max
Β· cases CharP.exists (R β§Έ M) with
| intro r hr =>
... | 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 161 | 169 | theorem of_algebraRat [Algebra β R] : β I : Ideal R, I β β€ β CharZero (R β§Έ I) := by |
intro I hI
constructor
intro a b h_ab
contrapose! hI
-- `βa - βb` is a unit contained in `I`, which contradicts `I β β€`.
refine I.eq_top_of_isUnit_mem ?_ (IsUnit.map (algebraMap β R) (IsUnit.mk0 (a - b : β) ?_))
Β· simpa only [β Ideal.Quotient.eq_zero_iff_mem, map_sub, sub_eq_zero, map_natCast]
simpa on... | 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 178 | 189 | theorem PNat.isUnit_natCast [h : Fact (β I : Ideal R, I β β€ β CharZero (R β§Έ I))]
(n : β+) : IsUnit (n : R) := by |
-- `n : R` is a unit iff `(n)` is not a proper ideal in `R`.
rw [β Ideal.span_singleton_eq_top]
-- So by contrapositive, we should show the quotient does not have characteristic zero.
apply not_imp_comm.mp (h.elim (Ideal.span {βn}))
intro h_char_zero
-- In particular, the image of `n` in the quotient shoul... | 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 204 | 209 | theorem pnatCast_one [Fact (β I : Ideal R, I β β€ β CharZero (R β§Έ I))] : ((1 : β+) : RΛ£) = 1 := by |
apply Units.ext
rw [Units.val_one]
change ((PNat.isUnit_natCast (R := R) 1).unit : R) = 1
rw [IsUnit.unit_spec (PNat.isUnit_natCast 1)]
rw [PNat.one_coe, Nat.cast_one]
| 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 214 | 217 | theorem pnatCast_eq_natCast [Fact (β I : Ideal R, I β β€ β CharZero (R β§Έ I))] (n : β+) :
((n : RΛ£) : R) = βn := by |
change ((PNat.isUnit_natCast (R := R) n).unit : R) = βn
simp only [IsUnit.unit_spec]
| 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 250 | 260 | theorem of_not_mixedCharZero [CharZero R] (h : β p > 0, Β¬MixedCharZero R p) :
β I : Ideal R, I β β€ β CharZero (R β§Έ I) := by |
intro I hI_ne_top
suffices CharP (R β§Έ I) 0 from CharP.charP_to_charZero _
cases CharP.exists (R β§Έ I) with
| intro p hp =>
cases p with
| zero => exact hp
| succ p =>
have h_mixed : MixedCharZero R p.succ := β¨β¨I, β¨hI_ne_top, hpβ©β©β©
exact absurd h_mixed (h p.succ p.succ_pos)
| 1,899 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : β) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : β I : Ideal R, I β β€ β CharZero (R β§Έ I)) :
β p > 0, Β¬MixedCharZero R p := by |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with β¨I, hI_ne_top, hI_pβ©
replace hI_zero : CharP (R β§Έ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R β§Έ I) hI_p hI_zero) (ne_of_gt p_pos)
| 1,899 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import rin... | Mathlib/RingTheory/Filtration.lean | 67 | 71 | theorem pow_smul_le (i j : β) : I ^ i β’ F.N j β€ F.N (i + j) := by |
induction' i with _ ih
Β· simp
Β· rw [pow_succ', mul_smul, add_assoc, add_comm 1, β add_assoc]
exact (smul_mono_right _ ih).trans (F.smul_le _)
| 1,900 |
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