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.NormedSpace.OperatorNorm.Basic
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special cases about linear (as opposed to semilinear) maps
variable {π πβ πβ E Eβ F Fβ G Gβ π : Type*}
... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 49 | 53 | theorem nnnorm_def (f : E βSL[Οββ] F) : βfββ = sInf { c | β x, βf xββ β€ c * βxββ } := by |
ext
rw [NNReal.coe_sInf, coe_nnnorm, norm_def, NNReal.coe_image]
simp_rw [β NNReal.coe_le_coe, NNReal.coe_mul, coe_nnnorm, mem_setOf_eq, NNReal.coe_mk,
exists_prop]
| 1,590 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special cases about linear (as opposed to semilinear) maps
variable {π πβ πβ E Eβ F Fβ G Gβ π : Type*}
... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 100 | 101 | theorem isLeast_opNNNorm : IsLeast {C : ββ₯0 | β x, βf xββ β€ C * βxββ} βfββ := by |
simpa only [β opNNNorm_le_iff] using isLeast_Ici
| 1,590 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 70 | 86 | theorem tendsto_of_tendsto_pointwise_of_cauchySeq {f : β β E' βSL[Οββ] F} {g : E' βSL[Οββ] F}
(hg : Tendsto (fun n x => f n x) atTop (π g)) (hf : CauchySeq f) : Tendsto f atTop (π g) := by |
/- Since `f` is a Cauchy sequence, there exists `b β 0` such that `βf n - f mβ β€ b N` for any
`m, n β₯ N`. -/
rcases cauchySeq_iff_le_tendsto_0.1 hf with β¨b, hbβ, hfb, hb_limβ©
-- Since `b β 0`, it suffices to show that `βf n x - g xβ β€ b n * βxβ` for all `n` and `x`.
suffices β n x, βf n x - g xβ β€ b n * βx... | 1,591 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
suppress_compilation
open Bornology Metric Set Real
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special cases about linear (as o... | Mathlib/Analysis/NormedSpace/OperatorNorm/Completeness.lean | 246 | 263 | theorem opNorm_extend_le :
βf.extend e h_dense (uniformEmbedding_of_bound _ h_e).toUniformInducingβ β€ N * βfβ := by |
-- Add `opNorm_le_of_dense`?
refine opNorm_le_bound _ ?_ (isClosed_property h_dense (isClosed_le ?_ ?_) fun x β¦ ?_)
Β· cases le_total 0 N with
| inl hN => exact mul_nonneg hN (norm_nonneg _)
| inr hN =>
have : Unique E := β¨β¨0β©, fun x β¦ norm_le_zero_iff.mp <|
(h_e x).trans (mul_nonpos_of_nonp... | 1,591 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 62 | 65 | theorem IsConformalMap.smul (hf : IsConformalMap f) {c : R} (hc : c β 0) :
IsConformalMap (c β’ f) := by |
rcases hf with β¨c', hc', li, rflβ©
exact β¨c * c', mul_ne_zero hc hc', li, smul_smul _ _ _β©
| 1,592 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 84 | 89 | theorem comp (hg : IsConformalMap g) (hf : IsConformalMap f) : IsConformalMap (g.comp f) := by |
rcases hf with β¨cf, hcf, lif, rflβ©
rcases hg with β¨cg, hcg, lig, rflβ©
refine β¨cg * cf, mul_ne_zero hcg hcf, lig.comp lif, ?_β©
rw [smul_comp, comp_smul, mul_smul]
rfl
| 1,592 |
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open Function LinearIsometry ContinuousLinearMap
def IsConf... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 97 | 100 | theorem ne_zero [Nontrivial M'] {f' : M' βL[R] N} (hf' : IsConformalMap f') : f' β 0 := by |
rintro rfl
rcases exists_ne (0 : M') with β¨a, haβ©
exact ha (hf'.injective rfl)
| 1,592 |
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
variable {π E : Type*}
namespace LinearMap
variable (π)
section Seminormed
variable [NormedDivisionRing π] [SeminormedAddCommGroup E] [Module π E] [BoundedSMu... | Mathlib/Analysis/NormedSpace/Span.lean | 36 | 39 | theorem toSpanSingleton_homothety (x : E) (c : π) :
βLinearMap.toSpanSingleton π E x cβ = βxβ * βcβ := by |
rw [mul_comm]
exact norm_smul _ _
| 1,593 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 42 | 46 | theorem bound_of_shell [RingHomIsometric Οββ] (f : E βββ[Οββ] F) {Ξ΅ C : β} (Ξ΅_pos : 0 < Ξ΅) {c : π}
(hc : 1 < βcβ) (hf : β x, Ξ΅ / βcβ β€ βxβ β βxβ < Ξ΅ β βf xβ β€ C * βxβ) (x : E) :
βf xβ β€ C * βxβ := by |
by_cases hx : x = 0; Β· simp [hx]
exact SemilinearMapClass.bound_of_shell_semi_normed f Ξ΅_pos hc hf (norm_ne_zero_iff.2 hx)
| 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 52 | 64 | theorem bound_of_ball_bound {r : β} (r_pos : 0 < r) (c : β) (f : E ββ[π] Fβ)
(h : β z β Metric.ball (0 : E) r, βf zβ β€ c) : β C, β z : E, βf zβ β€ C * βzβ := by |
cases' @NontriviallyNormedField.non_trivial π _ with k hk
use c * (βkβ / r)
intro z
refine bound_of_shell _ r_pos hk (fun x hko hxo => ?_) _
calc
βf xβ β€ c := h _ (mem_ball_zero_iff.mpr hxo)
_ β€ c * (βxβ * βkβ / r) := le_mul_of_one_le_right ?_ ?_
_ = _ := by ring
Β· exact le_trans (norm_nonneg ... | 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 67 | 87 | theorem antilipschitz_of_comap_nhds_le [h : RingHomIsometric Οββ] (f : E βββ[Οββ] F)
(hf : (π 0).comap f β€ π 0) : β K, AntilipschitzWith K f := by |
rcases ((nhds_basis_ball.comap _).le_basis_iff nhds_basis_ball).1 hf 1 one_pos with β¨Ξ΅, Ξ΅0, hΞ΅β©
simp only [Set.subset_def, Set.mem_preimage, mem_ball_zero_iff] at hΞ΅
lift Ξ΅ to ββ₯0 using Ξ΅0.le
rcases NormedField.exists_one_lt_norm π with β¨c, hcβ©
refine β¨Ξ΅β»ΒΉ * βcββ, AddMonoidHomClass.antilipschitz_of_bound f ... | 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 99 | 107 | theorem opNorm_zero_iff [RingHomIsometric Οββ] : βfβ = 0 β f = 0 :=
Iff.intro
(fun hn => ContinuousLinearMap.ext fun x => norm_le_zero_iff.1
(calc
_ β€ βfβ * βxβ := le_opNorm _ _
_ = _ := by | rw [hn, zero_mul]))
(by
rintro rfl
exact opNorm_zero)
| 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 114 | 117 | theorem norm_id [Nontrivial E] : βid π Eβ = 1 := by |
refine norm_id_of_nontrivial_seminorm ?_
obtain β¨x, hxβ© := exists_ne (0 : E)
exact β¨x, ne_of_gt (norm_pos_iff.2 hx)β©
| 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NNNorm
import Mathlib.Analysis.NormedSpace.Span
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `β` subscript variables are for special... | Mathlib/Analysis/NormedSpace/OperatorNorm/NormedSpace.lean | 140 | 146 | theorem homothety_norm [RingHomIsometric Οββ] [Nontrivial E] (f : E βSL[Οββ] F) {a : β}
(hf : β x, βf xβ = a * βxβ) : βfβ = a := by |
obtain β¨x, hxβ© : β x : E, x β 0 := exists_ne 0
rw [β norm_pos_iff] at hx
have ha : 0 β€ a := by simpa only [hf, hx, mul_nonneg_iff_of_pos_right] using norm_nonneg (f x)
apply le_antisymm (f.opNorm_le_bound ha fun y => le_of_eq (hf y))
simpa only [hf, hx, mul_le_mul_right] using f.le_opNorm x
| 1,594 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
suppress_compilation
set_option linter.uppercaseLean3 false
open Metric
open scoped Classical NNReal Topology Uniformity
variable {π E : Type*} [NontriviallyNormedField π]
section SemiNormed
variable [SeminormedAddCommGroup E] [NormedSpace π E]
... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 226 | 231 | theorem norm_toSpanSingleton (x : E) : βtoSpanSingleton π xβ = βxβ := by |
refine opNorm_eq_of_bounds (norm_nonneg _) (fun x => ?_) fun N _ h => ?_
Β· rw [toSpanSingleton_apply, norm_smul, mul_comm]
Β· specialize h 1
rw [toSpanSingleton_apply, norm_smul, mul_comm] at h
exact (mul_le_mul_right (by simp)).mp h
| 1,595 |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
suppress_compilation
set_option linter.uppercaseLean3 false
open Metric
open scoped Classical NNReal Topology Uniformity
variable {π E : Type*} [NontriviallyNormedField π]
section SemiNormed
variable [SeminormedAddCommGroup E] [NormedSpace π E]
... | Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 243 | 246 | theorem opNorm_lsmul_le : β(lsmul π π' : π' βL[π] E βL[π] E)β β€ 1 := by |
refine ContinuousLinearMap.opNorm_le_bound _ zero_le_one fun x => ?_
simp_rw [one_mul]
exact opNorm_lsmul_apply_le _
| 1,595 |
import Mathlib.Analysis.BoxIntegral.Partition.Split
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.box_integral.partition.additive from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Classical
open Function Set
namespace B... | Mathlib/Analysis/BoxIntegral/Partition/Additive.lean | 113 | 115 | theorem map_split_add (f : ΞΉ βα΅α΅[Iβ] M) (hI : βI β€ Iβ) (i : ΞΉ) (x : β) :
(I.splitLower i x).elim' 0 f + (I.splitUpper i x).elim' 0 f = f I := by |
rw [β f.sum_partition_boxes hI (isPartitionSplit I i x), sum_split_boxes]
| 1,596 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 89 | 101 | theorem splitMul_injective_of_clm_mul_injective
(h : Function.Injective (mul π A)) :
Function.Injective (splitMul π A) := by |
rw [injective_iff_map_eq_zero]
intro x hx
induction x
rw [map_add] at hx
simp only [splitMul_apply, fst_inl, snd_inl, map_zero, add_zero, fst_inr, snd_inr,
zero_add, Prod.mk_add_mk, Prod.mk_eq_zero] at hx
obtain β¨rfl, hxβ© := hx
simp only [map_zero, zero_add, inl_zero] at hx β’
rw [β map_zero (mul π... | 1,597 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 139 | 141 | theorem norm_eq_sup (x : Unitization π A) :
βxβ = βx.fstβ β βalgebraMap π (A βL[π] A) x.fst + mul π A x.sndβ := by |
rw [norm_def, splitMul_apply, Prod.norm_def, sup_eq_max]
| 1,597 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 149 | 165 | theorem lipschitzWith_addEquiv :
LipschitzWith 2 (Unitization.addEquiv π A) := by |
rw [β Real.toNNReal_ofNat]
refine AddMonoidHomClass.lipschitz_of_bound (Unitization.addEquiv π A) 2 fun x => ?_
rw [norm_eq_sup, Prod.norm_def]
refine max_le ?_ ?_
Β· rw [sup_eq_max, mul_max_of_nonneg _ _ (zero_le_two : (0 : β) β€ 2)]
exact le_max_of_le_left ((le_add_of_nonneg_left (norm_nonneg _)).trans_... | 1,597 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 167 | 180 | theorem antilipschitzWith_addEquiv :
AntilipschitzWith 2 (addEquiv π A) := by |
refine AddMonoidHomClass.antilipschitz_of_bound (addEquiv π A) fun x => ?_
rw [norm_eq_sup, Prod.norm_def, NNReal.coe_two]
refine max_le ?_ ?_
Β· rw [mul_max_of_nonneg _ _ (zero_le_two : (0 : β) β€ 2)]
exact le_max_of_le_left ((le_add_of_nonneg_left (norm_nonneg _)).trans_eq (two_mul _).symm)
Β· nontrivial... | 1,597 |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (π A : Type*) [NontriviallyNormedField π] [NonUnitalNormedRing A]
variable [NormedSpace π A] [IsScalarTower π A A] [SMulCommClass π A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 185 | 190 | theorem uniformity_eq_aux :
π€[instUniformSpaceProd.comap <| addEquiv π A] = π€ (Unitization π A) := by |
have key : UniformInducing (addEquiv π A) :=
antilipschitzWith_addEquiv.uniformInducing lipschitzWith_addEquiv.uniformContinuous
rw [β key.comap_uniformity]
rfl
| 1,597 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 115 | 118 | theorem fst : IsBoundedLinearMap π fun x : E Γ F => x.1 := by |
refine (LinearMap.fst π E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_left _ _
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 121 | 124 | theorem snd : IsBoundedLinearMap π fun x : E Γ F => x.2 := by |
refine (LinearMap.snd π E F).isLinear.with_bound 1 fun x => ?_
rw [one_mul]
exact le_max_right _ _
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 139 | 141 | theorem neg (hf : IsBoundedLinearMap π f) : IsBoundedLinearMap π fun e => -f e := by |
rw [show (fun e => -f e) = fun e => (-1 : π) β’ f e by funext; simp]
exact smul (-1) hf
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 144 | 151 | theorem add (hf : IsBoundedLinearMap π f) (hg : IsBoundedLinearMap π g) :
IsBoundedLinearMap π fun e => f e + g e :=
let β¨hlf, Mf, _, hMfβ© := hf
let β¨hlg, Mg, _, hMgβ© := hg
(hlf.mk' _ + hlg.mk' _).isLinear.with_bound (Mf + Mg) fun x =>
calc
βf x + g xβ β€ Mf * βxβ + Mg * βxβ := norm_add_le_of_le (... | rw [add_mul]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 155 | 156 | theorem sub (hf : IsBoundedLinearMap π f) (hg : IsBoundedLinearMap π g) :
IsBoundedLinearMap π fun e => f e - g e := by | simpa [sub_eq_add_neg] using add hf (neg hg)
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 217 | 231 | theorem isBoundedLinearMap_prod_multilinear {E : ΞΉ β Type*} [β i, NormedAddCommGroup (E i)]
[β i, NormedSpace π (E i)] :
IsBoundedLinearMap π fun p : ContinuousMultilinearMap π E F Γ ContinuousMultilinearMap π E G =>
p.1.prod p.2 where
map_add pβ pβ := by | ext : 1; rfl
map_smul c p := by ext : 1; rfl
bound := by
refine β¨1, zero_lt_one, fun p β¦ ?_β©
rw [one_mul]
apply ContinuousMultilinearMap.opNorm_le_bound _ (norm_nonneg _) _
intro m
rw [ContinuousMultilinearMap.prod_apply, norm_prod_le_iff]
constructor
Β· exact (p.1.le_opNorm m).trans (mu... | 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 236 | 254 | theorem isBoundedLinearMap_continuousMultilinearMap_comp_linear (g : G βL[π] E) :
IsBoundedLinearMap π fun f : ContinuousMultilinearMap π (fun _ : ΞΉ => E) F =>
f.compContinuousLinearMap fun _ => g := by |
refine
IsLinearMap.with_bound
β¨fun fβ fβ => by ext; rfl,
fun c f => by ext; rflβ©
(βgβ ^ Fintype.card ΞΉ) fun f => ?_
apply ContinuousMultilinearMap.opNorm_le_bound _ _ _
Β· apply_rules [mul_nonneg, pow_nonneg, norm_nonneg]
intro m
calc
βf (g β m)β β€ βfβ * β i, βg (m i)β := f.le_opNo... | 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 285 | 286 | theorem map_addβ (f : M βSL[Οββ] F βSL[Οββ] G') (x x' : M) (y : F) :
f (x + x') y = f x y + f x' y := by | rw [f.map_add, add_apply]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 289 | 290 | theorem map_zeroβ (f : M βSL[Οββ] F βSL[Οββ] G') (y : F) : f 0 y = 0 := by |
rw [f.map_zero, zero_apply]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 293 | 294 | theorem map_smulβββ (f : M βSL[Οββ] F βSL[Οββ] G') (c : R) (x : M) (y : F) :
f (c β’ x) y = Οββ c β’ f x y := by | rw [f.map_smulββ, smul_apply]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 303 | 304 | theorem map_subβ (f : M βSL[Οββ] F βSL[Οββ] G') (x x' : M) (y : F) :
f (x - x') y = f x y - f x' y := by | rw [f.map_sub, sub_apply]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 307 | 308 | theorem map_negβ (f : M βSL[Οββ] F βSL[Οββ] G') (x : M) (y : F) : f (-x) y = -f x y := by |
rw [f.map_neg, neg_apply]
| 1,598 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 313 | 314 | theorem map_smulβ (f : E βL[π] F βL[π] G) (c : π) (x : E) (y : F) : f (c β’ x) y = c β’ f x y := by |
rw [f.map_smul, smul_apply]
| 1,598 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.calculus.fderiv.linear from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topolo... | Mathlib/Analysis/Calculus/FDeriv/Linear.lean | 136 | 139 | theorem IsBoundedLinearMap.fderivWithin (h : IsBoundedLinearMap π f)
(hxs : UniqueDiffWithinAt π s x) : fderivWithin π f s x = h.toContinuousLinearMap := by |
rw [DifferentiableAt.fderivWithin h.differentiableAt hxs]
exact h.fderiv
| 1,599 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 346 | 350 | theorem HasStrictFDerivAt.sum (h : β i β u, HasStrictFDerivAt (A i) (A' i) x) :
HasStrictFDerivAt (fun y => β i β u, A i y) (β i β u, A' i) x := by |
dsimp [HasStrictFDerivAt] at *
convert IsLittleO.sum h
simp [Finset.sum_sub_distrib, ContinuousLinearMap.sum_apply]
| 1,600 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 353 | 357 | theorem HasFDerivAtFilter.sum (h : β i β u, HasFDerivAtFilter (A i) (A' i) x L) :
HasFDerivAtFilter (fun y => β i β u, A i y) (β i β u, A' i) x L := by |
simp only [hasFDerivAtFilter_iff_isLittleO] at *
convert IsLittleO.sum h
simp [ContinuousLinearMap.sum_apply]
| 1,600 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 488 | 489 | theorem fderiv_neg : fderiv π (fun y => -f y) x = -fderiv π f x := by |
simp only [β fderivWithin_univ, fderivWithin_neg uniqueDiffWithinAt_univ]
| 1,600 |
import Mathlib.Analysis.Calculus.FDeriv.Add
variable {π ΞΉ : Type*} [DecidableEq ΞΉ] [Fintype ΞΉ] [NontriviallyNormedField π]
variable {E : ΞΉ β Type*} [β i, NormedAddCommGroup (E i)] [β i, NormedSpace π (E i)]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
@[fun_prop]
| Mathlib/Analysis/Calculus/FDeriv/Pi.lean | 17 | 29 | theorem hasFDerivAt_update (x : β i, E i) {i : ΞΉ} (y : E i) :
HasFDerivAt (Function.update x i) (.pi (Pi.single i (.id π (E i)))) y := by |
set l := (ContinuousLinearMap.pi (Pi.single i (.id π (E i))))
have update_eq : Function.update x i = (fun _ β¦ x) + l β (Β· - x i) := by
ext t j
dsimp [l, Pi.single, Function.update]
split_ifs with hji
Β· subst hji
simp
Β· simp
rw [update_eq]
convert (hasFDerivAt_const _ _).add (l.hasFDe... | 1,601 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 73 | 82 | theorem conformalAt_iff_isConformalMap_fderiv {f : X β Y} {x : X} :
ConformalAt f x β IsConformalMap (fderiv β f x) := by |
constructor
Β· rintro β¨f', hf, hf'β©
rwa [hf.fderiv]
Β· intro H
by_cases h : DifferentiableAt β f x
Β· exact β¨fderiv β f x, h.hasFDerivAt, Hβ©
Β· nontriviality X
exact absurd (fderiv_zero_of_not_differentiableAt h) H.ne_zero
| 1,602 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 98 | 102 | theorem comp {f : X β Y} {g : Y β Z} (x : X) (hg : ConformalAt g (f x)) (hf : ConformalAt f x) :
ConformalAt (g β f) x := by |
rcases hf with β¨f', hfβ, cfβ©
rcases hg with β¨g', hgβ, cgβ©
exact β¨g'.comp f', hgβ.comp x hfβ, cg.comp cfβ©
| 1,602 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 400 | 403 | theorem hasStrictFDerivAt_pi' :
HasStrictFDerivAt Ξ¦ Ξ¦' x β β i, HasStrictFDerivAt (fun x => Ξ¦ x i) ((proj i).comp Ξ¦') x := by |
simp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]
exact isLittleO_pi
| 1,603 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 411 | 417 | theorem hasStrictFDerivAt_apply (i : ΞΉ) (f : β i, F' i) :
HasStrictFDerivAt (π:=π) (fun f : β i, F' i => f i) (proj i) f := by |
let id' := ContinuousLinearMap.id π (β i, F' i)
have h := ((hasStrictFDerivAt_pi'
(Ξ¦ := fun (f : β i, F' i) (i' : ΞΉ) => f i') (Ξ¦':=id') (x:=f))).1
have h' : comp (proj i) id' = proj i := by rfl
rw [β h']; apply h; apply hasStrictFDerivAt_id
| 1,603 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 427 | 431 | theorem hasFDerivAtFilter_pi' :
HasFDerivAtFilter Ξ¦ Ξ¦' x L β
β i, HasFDerivAtFilter (fun x => Ξ¦ x i) ((proj i).comp Ξ¦') x L := by |
simp only [hasFDerivAtFilter_iff_isLittleO, ContinuousLinearMap.coe_pi]
exact isLittleO_pi
| 1,603 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 451 | 454 | theorem hasFDerivAt_apply (i : ΞΉ) (f : β i, F' i) :
HasFDerivAt (π:=π) (fun f : β i, F' i => f i) (proj i) f := by |
apply HasStrictFDerivAt.hasFDerivAt
apply hasStrictFDerivAt_apply
| 1,603 |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.prod from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal ... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 474 | 480 | theorem hasFDerivWithinAt_apply (i : ΞΉ) (f : β i, F' i) (s' : Set (β i, F' i)) :
HasFDerivWithinAt (π:=π) (fun f : β i, F' i => f i) (proj i) s' f := by |
let id' := ContinuousLinearMap.id π (β i, F' i)
have h := ((hasFDerivWithinAt_pi'
(Ξ¦ := fun (f : β i, F' i) (i' : ΞΉ) => f i') (Ξ¦':=id') (x:=f) (s:=s'))).1
have h' : comp (proj i) id' = proj i := by rfl
rw [β h']; apply h; apply hasFDerivWithinAt_id
| 1,603 |
import Mathlib.Analysis.Calculus.FDeriv.Prod
#align_import analysis.calculus.fderiv.bilinear from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Asymptotics ENNReal
noncomputable section
... | Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean | 51 | 74 | theorem IsBoundedBilinearMap.hasStrictFDerivAt (h : IsBoundedBilinearMap π b) (p : E Γ F) :
HasStrictFDerivAt b (h.deriv p) p := by |
simp only [HasStrictFDerivAt]
simp only [β map_add_left_nhds_zero (p, p), isLittleO_map]
set T := (E Γ F) Γ E Γ F
calc
_ = fun x β¦ h.deriv (x.1 - x.2) (x.2.1, x.1.2) := by
ext β¨β¨xβ, yββ©, β¨xβ, yββ©β©
rcases p with β¨x, yβ©
simp only [map_sub, deriv_apply, Function.comp_apply, Prod.mk_add_mk, h... | 1,604 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 224 | 227 | theorem fderivWithin_continuousMultilinear_apply_const_apply (hxs : UniqueDiffWithinAt π s x)
(hc : DifferentiableWithinAt π c s x) (u : β i, M i) (m : E) :
(fderivWithin π (fun y β¦ (c y) u) s x) m = (fderivWithin π c s x) m u := by |
simp [fderivWithin_continuousMultilinear_apply_const hxs hc]
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 230 | 233 | theorem fderiv_continuousMultilinear_apply_const_apply (hc : DifferentiableAt π c x)
(u : β i, M i) (m : E) :
(fderiv π (fun y β¦ (c y) u) x) m = (fderiv π c x) m u := by |
simp [fderiv_continuousMultilinear_apply_const hc]
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 307 | 309 | theorem HasStrictFDerivAt.smul_const (hc : HasStrictFDerivAt c c' x) (f : F) :
HasStrictFDerivAt (fun y => c y β’ f) (c'.smulRight f) x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasStrictFDerivAt_const f x)
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 313 | 315 | theorem HasFDerivWithinAt.smul_const (hc : HasFDerivWithinAt c c' s x) (f : F) :
HasFDerivWithinAt (fun y => c y β’ f) (c'.smulRight f) s x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasFDerivWithinAt_const f x s)
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 319 | 321 | theorem HasFDerivAt.smul_const (hc : HasFDerivAt c c' x) (f : F) :
HasFDerivAt (fun y => c y β’ f) (c'.smulRight f) x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasFDerivAt_const f x)
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 376 | 380 | theorem HasStrictFDerivAt.mul (hc : HasStrictFDerivAt c c' x) (hd : HasStrictFDerivAt d d' x) :
HasStrictFDerivAt (fun y => c y * d y) (c x β’ d' + d x β’ c') x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 391 | 395 | theorem HasFDerivWithinAt.mul (hc : HasFDerivWithinAt c c' s x) (hd : HasFDerivWithinAt d d' s x) :
HasFDerivWithinAt (fun y => c y * d y) (c x β’ d' + d x β’ c') s x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 1,605 |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 405 | 409 | theorem HasFDerivAt.mul (hc : HasFDerivAt c c' x) (hd : HasFDerivAt d d' x) :
HasFDerivAt (fun y => c y * d y) (c x β’ d' + d x β’ c') x := by |
convert hc.mul' hd
ext z
apply mul_comm
| 1,605 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 95 | 101 | theorem comp_differentiableWithinAt_iff {f : G β E} {s : Set G} {x : G} :
DifferentiableWithinAt π (iso β f) s x β DifferentiableWithinAt π f s x := by |
refine
β¨fun H => ?_, fun H => iso.differentiable.differentiableAt.comp_differentiableWithinAt x Hβ©
have : DifferentiableWithinAt π (iso.symm β iso β f) s x :=
iso.symm.differentiable.differentiableAt.comp_differentiableWithinAt x H
rwa [β Function.comp.assoc iso.symm iso f, iso.symm_comp_self] at this
| 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 104 | 107 | theorem comp_differentiableAt_iff {f : G β E} {x : G} :
DifferentiableAt π (iso β f) x β DifferentiableAt π f x := by |
rw [β differentiableWithinAt_univ, β differentiableWithinAt_univ,
iso.comp_differentiableWithinAt_iff]
| 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 110 | 113 | theorem comp_differentiableOn_iff {f : G β E} {s : Set G} :
DifferentiableOn π (iso β f) s β DifferentiableOn π f s := by |
rw [DifferentiableOn, DifferentiableOn]
simp only [iso.comp_differentiableWithinAt_iff]
| 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 116 | 118 | theorem comp_differentiable_iff {f : G β E} : Differentiable π (iso β f) β Differentiable π f := by |
rw [β differentiableOn_univ, β differentiableOn_univ]
exact iso.comp_differentiableOn_iff
| 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 121 | 130 | theorem comp_hasFDerivWithinAt_iff {f : G β E} {s : Set G} {x : G} {f' : G βL[π] E} :
HasFDerivWithinAt (iso β f) ((iso : E βL[π] F).comp f') s x β HasFDerivWithinAt f f' s x := by |
refine β¨fun H => ?_, fun H => iso.hasFDerivAt.comp_hasFDerivWithinAt x Hβ©
have A : f = iso.symm β iso β f := by
rw [β Function.comp.assoc, iso.symm_comp_self]
rfl
have B : f' = (iso.symm : F βL[π] E).comp ((iso : E βL[π] F).comp f') := by
rw [β ContinuousLinearMap.comp_assoc, iso.coe_symm_comp_coe,... | 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 133 | 137 | theorem comp_hasStrictFDerivAt_iff {f : G β E} {x : G} {f' : G βL[π] E} :
HasStrictFDerivAt (iso β f) ((iso : E βL[π] F).comp f') x β HasStrictFDerivAt f f' x := by |
refine β¨fun H => ?_, fun H => iso.hasStrictFDerivAt.comp x Hβ©
convert iso.symm.hasStrictFDerivAt.comp x H using 1 <;>
ext z <;> apply (iso.symm_apply_apply _).symm
| 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 391 | 410 | theorem HasStrictFDerivAt.of_local_left_inverse {f : E β F} {f' : E βL[π] F} {g : F β E} {a : F}
(hg : ContinuousAt g a) (hf : HasStrictFDerivAt f (f' : E βL[π] F) (g a))
(hfg : βαΆ y in π a, f (g y) = y) : HasStrictFDerivAt g (f'.symm : F βL[π] E) a := by |
replace hg := hg.prod_map' hg
replace hfg := hfg.prod_mk_nhds hfg
have :
(fun p : F Γ F => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[π (a, a)] fun p : F Γ F =>
f' (g p.1 - g p.2) - (p.1 - p.2) := by
refine ((f'.symm : F βL[π] E).isBigO_comp _ _).congr (fun x => ?_) fun _ => rfl
simp
refine th... | 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 418 | 433 | theorem HasFDerivAt.of_local_left_inverse {f : E β F} {f' : E βL[π] F} {g : F β E} {a : F}
(hg : ContinuousAt g a) (hf : HasFDerivAt f (f' : E βL[π] F) (g a))
(hfg : βαΆ y in π a, f (g y) = y) : HasFDerivAt g (f'.symm : F βL[π] E) a := by |
have : (fun x : F => g x - g a - f'.symm (x - a)) =O[π a]
fun x : F => f' (g x - g a) - (x - a) := by
refine ((f'.symm : F βL[π] E).isBigO_comp _ _).congr (fun x => ?_) fun _ => rfl
simp
refine HasFDerivAtFilter.of_isLittleO <| this.trans_isLittleO ?_
clear this
refine ((hf.isLittleO.comp_tends... | 1,606 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 459 | 465 | theorem HasFDerivWithinAt.eventually_ne (h : HasFDerivWithinAt f f' s x)
(hf' : β C, β z, βzβ β€ C * βf' zβ) : βαΆ z in π[s \ {x}] x, f z β f x := by |
rw [nhdsWithin, diff_eq, β inf_principal, β inf_assoc, eventually_inf_principal]
have A : (fun z => z - x) =O[π[s] x] fun z => f' (z - x) :=
isBigO_iff.2 <| hf'.imp fun C hC => eventually_of_forall fun z => hC _
have : (fun z => f z - f x) ~[π[s] x] fun z => f' (z - x) := h.isLittleO.trans_isBigO A
simpa... | 1,606 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 196 | 199 | theorem HasFTaylorSeriesUpToOn.zero_eq' (h : HasFTaylorSeriesUpToOn n f p s) {x : E} (hx : x β s) :
p x 0 = (continuousMultilinearCurryFin0 π E F).symm (f x) := by |
rw [β h.zero_eq x hx]
exact (p x 0).uncurry0_curry0.symm
| 1,607 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 204 | 208 | theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
(hβ : β x β s, fβ x = f x) : HasFTaylorSeriesUpToOn n fβ p s := by |
refine β¨fun x hx => ?_, h.fderivWithin, h.contβ©
rw [hβ x hx]
exact h.zero_eq x hx
| 1,607 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 223 | 226 | theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
ContinuousOn f s := by |
have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
rwa [β (continuousMultilinearCurryFin0 π E F).symm.comp_continuousOn_iff]
| 1,607 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 229 | 237 | theorem hasFTaylorSeriesUpToOn_zero_iff :
HasFTaylorSeriesUpToOn 0 f p s β ContinuousOn f s β§ β x β s, (p x 0).uncurry0 = f x := by |
refine β¨fun H => β¨H.continuousOn, H.zero_eqβ©, fun H =>
β¨H.2, fun m hm => False.elim (not_le.2 hm bot_le), fun m hm β¦ ?_β©β©
obtain rfl : m = 0 := mod_cast hm.antisymm (zero_le _)
have : EqOn (p Β· 0) ((continuousMultilinearCurryFin0 π E F).symm β f) s := fun x hx β¦
(continuousMultilinearCurryFin0 π E F)... | 1,607 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 240 | 250 | theorem hasFTaylorSeriesUpToOn_top_iff :
HasFTaylorSeriesUpToOn β f p s β β n : β, HasFTaylorSeriesUpToOn n f p s := by |
constructor
Β· intro H n; exact H.of_le le_top
Β· intro H
constructor
Β· exact (H 0).zero_eq
Β· intro m _
apply (H m.succ).fderivWithin m (WithTop.coe_lt_coe.2 (lt_add_one m))
Β· intro m _
apply (H m).cont m le_rfl
| 1,607 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 120 | 123 | theorem coe_linearMapAt (e : Pretrivialization F (Ο F E)) [e.IsLinear R] (b : B) :
β(e.linearMapAt R b) = fun y => if b β e.baseSet then (e β¨b, yβ©).2 else 0 := by |
rw [Pretrivialization.linearMapAt]
split_ifs <;> rfl
| 1,608 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 126 | 128 | theorem coe_linearMapAt_of_mem (e : Pretrivialization F (Ο F E)) [e.IsLinear R] {b : B}
(hb : b β e.baseSet) : β(e.linearMapAt R b) = fun y => (e β¨b, yβ©).2 := by |
simp_rw [coe_linearMapAt, if_pos hb]
| 1,608 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 131 | 133 | theorem linearMapAt_apply (e : Pretrivialization F (Ο F E)) [e.IsLinear R] {b : B} (y : E b) :
e.linearMapAt R b y = if b β e.baseSet then (e β¨b, yβ©).2 else 0 := by |
rw [coe_linearMapAt]
| 1,608 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 151 | 154 | theorem symmβ_linearMapAt (e : Pretrivialization F (Ο F E)) [e.IsLinear R] {b : B}
(hb : b β e.baseSet) (y : E b) : e.symmβ R b (e.linearMapAt R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).left_inv y
| 1,608 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 157 | 160 | theorem linearMapAt_symmβ (e : Pretrivialization F (Ο F E)) [e.IsLinear R] {b : B}
(hb : b β e.baseSet) (y : F) : e.linearMapAt R b (e.symmβ R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).right_inv y
| 1,608 |
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
#align_import topology.vector_bundle.constructions from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Cl... | Mathlib/Topology/VectorBundle/Constructions.lean | 50 | 55 | theorem trivialization.coordChangeL (b : B) :
(trivialization B F).coordChangeL π (trivialization B F) b =
ContinuousLinearEquiv.refl π F := by |
ext v
rw [Trivialization.coordChangeL_apply']
exacts [rfl, β¨mem_univ _, mem_univ _β©]
| 1,609 |
import Mathlib.Topology.FiberBundle.Constructions
import Mathlib.Topology.VectorBundle.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
#align_import topology.vector_bundle.constructions from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Cl... | Mathlib/Topology/VectorBundle/Constructions.lean | 96 | 106 | theorem coordChangeL_prod [eβ.IsLinear π] [eβ'.IsLinear π] [eβ.IsLinear π] [eβ'.IsLinear π] β¦bβ¦
(hb : b β (eβ.prod eβ).baseSet β© (eβ'.prod eβ').baseSet) :
((eβ.prod eβ).coordChangeL π (eβ'.prod eβ') b : Fβ Γ Fβ βL[π] Fβ Γ Fβ) =
(eβ.coordChangeL π eβ' b : Fβ βL[π] Fβ).prodMap (eβ.coordChangeL π eβ... |
rw [ContinuousLinearMap.ext_iff, ContinuousLinearMap.coe_prodMap']
rintro β¨vβ, vββ©
show
(eβ.prod eβ).coordChangeL π (eβ'.prod eβ') b (vβ, vβ) =
(eβ.coordChangeL π eβ' b vβ, eβ.coordChangeL π eβ' b vβ)
rw [eβ.coordChangeL_apply eβ', eβ.coordChangeL_apply eβ', (eβ.prod eβ).coordChangeL_apply']
exa... | 1,609 |
import Mathlib.Topology.VectorBundle.Basic
#align_import topology.vector_bundle.hom from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
noncomputable section
open scoped Bundle
open Bundle Set ContinuousLinearMap
variable {πβ : Type*} [NontriviallyNormedField πβ] {πβ : Type*} [Non... | Mathlib/Topology/VectorBundle/Hom.lean | 92 | 112 | theorem continuousOn_continuousLinearMapCoordChange [VectorBundle πβ Fβ Eβ] [VectorBundle πβ Fβ Eβ]
[MemTrivializationAtlas eβ] [MemTrivializationAtlas eβ'] [MemTrivializationAtlas eβ]
[MemTrivializationAtlas eβ'] :
ContinuousOn (continuousLinearMapCoordChange Ο eβ eβ' eβ eβ')
(eβ.baseSet β© eβ.baseS... |
have hβ := (compSL Fβ Fβ Fβ Ο (RingHom.id πβ)).continuous
have hβ := (ContinuousLinearMap.flip (compSL Fβ Fβ Fβ (RingHom.id πβ) Ο)).continuous
have hβ := continuousOn_coordChange πβ eβ' eβ
have hβ := continuousOn_coordChange πβ eβ eβ'
refine ((hβ.comp_continuousOn (hβ.mono ?_)).clm_comp (hβ.comp_continuo... | 1,610 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 161 | 162 | theorem hasFDerivAtFilter_iff_hasDerivAtFilter {f' : π βL[π] F} :
HasFDerivAtFilter f f' x L β HasDerivAtFilter f (f' 1) x L := by | simp [HasDerivAtFilter]
| 1,611 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.basic from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical Topology Filter ENNReal ... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 201 | 203 | theorem hasStrictFDerivAt_iff_hasStrictDerivAt {f' : π βL[π] F} :
HasStrictFDerivAt f f' x β HasStrictDerivAt f (f' 1) x := by |
simp [HasStrictDerivAt, HasStrictFDerivAt]
| 1,611 |
import Mathlib.Analysis.Calculus.Deriv.Basic
#align_import analysis.calculus.deriv.support from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v
variable {π : Type u} [NontriviallyNormedField π]
variable {E : Type v} [NormedAddCommGroup E] [NormedSpace π E]
variable {f : ... | Mathlib/Analysis/Calculus/Deriv/Support.lean | 36 | 41 | theorem support_deriv_subset : support (deriv f) β tsupport f := by |
intro x
rw [β not_imp_not]
intro h2x
rw [not_mem_tsupport_iff_eventuallyEq] at h2x
exact nmem_support.mpr (h2x.deriv_eq.trans (deriv_const x 0))
| 1,612 |
import Mathlib.Analysis.Calculus.FDeriv.Pi
import Mathlib.Analysis.Calculus.Deriv.Basic
variable {π ΞΉ : Type*} [DecidableEq ΞΉ] [Fintype ΞΉ] [NontriviallyNormedField π]
| Mathlib/Analysis/Calculus/Deriv/Pi.lean | 15 | 22 | theorem hasDerivAt_update (x : ΞΉ β π) (i : ΞΉ) (y : π) :
HasDerivAt (Function.update x i) (Pi.single i (1 : π)) y := by |
convert (hasFDerivAt_update x y).hasDerivAt
ext z j
rw [Pi.single, Function.update_apply]
split_ifs with h
Β· simp [h]
Β· simp [Pi.single_eq_of_ne h]
| 1,613 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 74 | 77 | theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter gβ gβ' (h x) L')
(hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
HasDerivAtFilter (gβ β h) (h' β’ gβ') x L := by |
simpa using ((hg.restrictScalars π).comp x hh hL).hasDerivAtFilter
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 80 | 83 | theorem HasDerivAtFilter.scomp_of_eq (hg : HasDerivAtFilter gβ gβ' y L')
(hh : HasDerivAtFilter h h' x L) (hy : y = h x) (hL : Tendsto h L L') :
HasDerivAtFilter (gβ β h) (h' β’ gβ') x L := by |
rw [hy] at hg; exact hg.scomp x hh hL
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 90 | 93 | theorem HasDerivWithinAt.scomp_hasDerivAt_of_eq (hg : HasDerivWithinAt gβ gβ' s' y)
(hh : HasDerivAt h h' x) (hs : β x, h x β s') (hy : y = h x) :
HasDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp_hasDerivAt x hh hs
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 101 | 104 | theorem HasDerivWithinAt.scomp_of_eq (hg : HasDerivWithinAt gβ gβ' t' y)
(hh : HasDerivWithinAt h h' s x) (hst : MapsTo h s t') (hy : y = h x) :
HasDerivWithinAt (gβ β h) (h' β’ gβ') s x := by |
rw [hy] at hg; exact hg.scomp x hh hst
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 113 | 116 | theorem HasDerivAt.scomp_of_eq
(hg : HasDerivAt gβ gβ' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
HasDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp x hh
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 118 | 120 | theorem HasStrictDerivAt.scomp (hg : HasStrictDerivAt gβ gβ' (h x)) (hh : HasStrictDerivAt h h' x) :
HasStrictDerivAt (gβ β h) (h' β’ gβ') x := by |
simpa using ((hg.restrictScalars π).comp x hh).hasStrictDerivAt
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 123 | 126 | theorem HasStrictDerivAt.scomp_of_eq
(hg : HasStrictDerivAt gβ gβ' y) (hh : HasStrictDerivAt h h' x) (hy : y = h x) :
HasStrictDerivAt (gβ β h) (h' β’ gβ') x := by |
rw [hy] at hg; exact hg.scomp x hh
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 133 | 136 | theorem HasDerivAt.scomp_hasDerivWithinAt_of_eq (hg : HasDerivAt gβ gβ' y)
(hh : HasDerivWithinAt h h' s x) (hy : y = h x) :
HasDerivWithinAt (gβ β h) (h' β’ gβ') s x := by |
rw [hy] at hg; exact hg.scomp_hasDerivWithinAt x hh
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 357 | 361 | theorem HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq {t : Set F}
(hl : HasFDerivWithinAt l l' t y)
(hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) (hy : y = f x) :
HasDerivWithinAt (l β f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 368 | 371 | theorem HasFDerivAt.comp_hasDerivWithinAt_of_eq (hl : HasFDerivAt l l' y)
(hf : HasDerivWithinAt f f' s x) (hy : y = f x) :
HasDerivWithinAt (l β f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 382 | 385 | theorem HasFDerivAt.comp_hasDerivAt_of_eq
(hl : HasFDerivAt l l' y) (hf : HasDerivAt f f' x) (hy : y = f x) :
HasDerivAt (l β f) (l' f') x := by |
rw [hy] at hl; exact hl.comp_hasDerivAt x hf
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 393 | 396 | theorem HasStrictFDerivAt.comp_hasStrictDerivAt_of_eq (hl : HasStrictFDerivAt l l' y)
(hf : HasStrictDerivAt f f' x) (hy : y = f x) :
HasStrictDerivAt (l β f) (l' f') x := by |
rw [hy] at hl; exact hl.comp_hasStrictDerivAt x hf
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 404 | 408 | theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt π l t y)
(hf : DifferentiableWithinAt π f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x)
(hy : y = f x) :
derivWithin (l β f) s x = (fderivWithin π l t (f x) : F β E) (derivWithin f s x) := by |
rw [hy] at hl; exact fderivWithin.comp_derivWithin x hl hf hs hxs
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 415 | 418 | theorem fderiv.comp_deriv_of_eq (hl : DifferentiableAt π l y) (hf : DifferentiableAt π f x)
(hy : y = f x) :
deriv (l β f) x = (fderiv π l (f x) : F β E) (deriv f x) := by |
rw [hy] at hl; exact fderiv.comp_deriv x hl hf
| 1,614 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_import analysis.calculus.deriv.inverse from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
va... | Mathlib/Analysis/Calculus/Deriv/Inverse.lean | 112 | 117 | theorem not_differentiableWithinAt_of_local_left_inverse_hasDerivWithinAt_zero {f g : π β π} {a : π}
{s t : Set π} (ha : a β s) (hsu : UniqueDiffWithinAt π s a) (hf : HasDerivWithinAt f 0 t (g a))
(hst : MapsTo g s t) (hfg : f β g =αΆ [π[s] a] id) : Β¬DifferentiableWithinAt π g s a := by |
intro hg
have := (hf.comp a hg.hasDerivWithinAt hst).congr_of_eventuallyEq_of_mem hfg.symm ha
simpa using hsu.eq_deriv _ this (hasDerivWithinAt_id _ _)
| 1,615 |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_import analysis.calculus.deriv.inverse from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
va... | Mathlib/Analysis/Calculus/Deriv/Inverse.lean | 120 | 124 | theorem not_differentiableAt_of_local_left_inverse_hasDerivAt_zero {f g : π β π} {a : π}
(hf : HasDerivAt f 0 (g a)) (hfg : f β g =αΆ [π a] id) : Β¬DifferentiableAt π g a := by |
intro hg
have := (hf.comp a hg.hasDerivAt).congr_of_eventuallyEq hfg.symm
simpa using this.unique (hasDerivAt_id a)
| 1,615 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 62 | 70 | theorem iteratedFDerivWithin_zero_fun (hs : UniqueDiffOn π s) (hx : x β s) {i : β} :
iteratedFDerivWithin π i (fun _ : E β¦ (0 : F)) s x = 0 := by |
induction i generalizing x with
| zero => ext; simp
| succ i IH =>
ext m
rw [iteratedFDerivWithin_succ_apply_left, fderivWithin_congr (fun _ β¦ IH) (IH hx)]
rw [fderivWithin_const_apply _ (hs x hx)]
rfl
| 1,616 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 86 | 91 | theorem contDiff_const {c : F} : ContDiff π n fun _ : E => c := by |
suffices h : ContDiff π β fun _ : E => c from h.of_le le_top
rw [contDiff_top_iff_fderiv]
refine β¨differentiable_const c, ?_β©
rw [fderiv_const]
exact contDiff_zero_fun
| 1,616 |
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