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 |
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import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F F' G 𝕜 : Type*} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F']
[NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α}
local infixr:25 " →ₛ " => SimpleFunc
open Finset
section FinMeasAdditive
def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) :
Prop :=
∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t
#align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive
namespace FinMeasAdditive
variable {β : Type*} [AddCommMonoid β] {T T' : Set α → β}
theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp
#align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 105 | 109 | theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T') := by |
intro s t hs ht hμs hμt hst
simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]
abel
| 2,117 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F F' G 𝕜 : Type*} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F']
[NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α}
local infixr:25 " →ₛ " => SimpleFunc
open Finset
section FinMeasAdditive
def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) :
Prop :=
∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t
#align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive
namespace FinMeasAdditive
variable {β : Type*} [AddCommMonoid β] {T T' : Set α → β}
theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp
#align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero
theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T') := by
intro s t hs ht hμs hμt hst
simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]
abel
#align measure_theory.fin_meas_additive.add MeasureTheory.FinMeasAdditive.add
theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
FinMeasAdditive μ fun s => c • T s := fun s t hs ht hμs hμt hst => by
simp [hT s t hs ht hμs hμt hst]
#align measure_theory.fin_meas_additive.smul MeasureTheory.FinMeasAdditive.smul
theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞)
(hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T := fun s t hs ht hμ's hμ't hst =>
hT s t hs ht (mt (h s hs) hμ's) (mt (h t ht) hμ't) hst
#align measure_theory.fin_meas_additive.of_eq_top_imp_eq_top MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
| 2,117 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F F' G 𝕜 : Type*} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F']
[NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α}
local infixr:25 " →ₛ " => SimpleFunc
open Finset
section FinMeasAdditive
def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) :
Prop :=
∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t
#align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive
namespace FinMeasAdditive
variable {β : Type*} [AddCommMonoid β] {T T' : Set α → β}
theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp
#align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero
theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T') := by
intro s t hs ht hμs hμt hst
simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]
abel
#align measure_theory.fin_meas_additive.add MeasureTheory.FinMeasAdditive.add
theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
FinMeasAdditive μ fun s => c • T s := fun s t hs ht hμs hμt hst => by
simp [hT s t hs ht hμs hμt hst]
#align measure_theory.fin_meas_additive.smul MeasureTheory.FinMeasAdditive.smul
theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞)
(hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T := fun s t hs ht hμ's hμ't hst =>
hT s t hs ht (mt (h s hs) hμ's) (mt (h t ht) hμ't) hst
#align measure_theory.fin_meas_additive.of_eq_top_imp_eq_top MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top
theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
#align measure_theory.fin_meas_additive.of_smul_measure MeasureTheory.FinMeasAdditive.of_smul_measure
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 130 | 135 | theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
FinMeasAdditive (c • μ) T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top]
simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff]
exact Or.inl hμs
| 2,117 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F F' G 𝕜 : Type*} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F']
[NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α}
local infixr:25 " →ₛ " => SimpleFunc
open Finset
section FinMeasAdditive
def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) :
Prop :=
∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t
#align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive
namespace FinMeasAdditive
variable {β : Type*} [AddCommMonoid β] {T T' : Set α → β}
theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp
#align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero
theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T') := by
intro s t hs ht hμs hμt hst
simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]
abel
#align measure_theory.fin_meas_additive.add MeasureTheory.FinMeasAdditive.add
theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
FinMeasAdditive μ fun s => c • T s := fun s t hs ht hμs hμt hst => by
simp [hT s t hs ht hμs hμt hst]
#align measure_theory.fin_meas_additive.smul MeasureTheory.FinMeasAdditive.smul
theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞)
(hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T := fun s t hs ht hμ's hμ't hst =>
hT s t hs ht (mt (h s hs) hμ's) (mt (h t ht) hμ't) hst
#align measure_theory.fin_meas_additive.of_eq_top_imp_eq_top MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top
theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
#align measure_theory.fin_meas_additive.of_smul_measure MeasureTheory.FinMeasAdditive.of_smul_measure
theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
FinMeasAdditive (c • μ) T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top]
simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff]
exact Or.inl hμs
#align measure_theory.fin_meas_additive.smul_measure MeasureTheory.FinMeasAdditive.smul_measure
theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) :
FinMeasAdditive (c • μ) T ↔ FinMeasAdditive μ T :=
⟨fun hT => of_smul_measure c hc_ne_top hT, fun hT => smul_measure c hc_ne_zero hT⟩
#align measure_theory.fin_meas_additive.smul_measure_iff MeasureTheory.FinMeasAdditive.smul_measure_iff
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 143 | 149 | theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) :
T ∅ = 0 := by |
have h_empty : μ ∅ ≠ ∞ := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne
specialize hT ∅ ∅ MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty ∅)
rw [Set.union_empty] at hT
nth_rw 1 [← add_zero (T ∅)] at hT
exact (add_left_cancel hT).symm
| 2,117 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F F' G 𝕜 : Type*} {p : ℝ≥0∞} [NormedAddCommGroup E] [NormedSpace ℝ E]
[NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup F'] [NormedSpace ℝ F']
[NormedAddCommGroup G] {m : MeasurableSpace α} {μ : Measure α}
local infixr:25 " →ₛ " => SimpleFunc
open Finset
section FinMeasAdditive
def FinMeasAdditive {β} [AddMonoid β] {_ : MeasurableSpace α} (μ : Measure α) (T : Set α → β) :
Prop :=
∀ s t, MeasurableSet s → MeasurableSet t → μ s ≠ ∞ → μ t ≠ ∞ → s ∩ t = ∅ → T (s ∪ t) = T s + T t
#align measure_theory.fin_meas_additive MeasureTheory.FinMeasAdditive
namespace FinMeasAdditive
variable {β : Type*} [AddCommMonoid β] {T T' : Set α → β}
theorem zero : FinMeasAdditive μ (0 : Set α → β) := fun s t _ _ _ _ _ => by simp
#align measure_theory.fin_meas_additive.zero MeasureTheory.FinMeasAdditive.zero
theorem add (hT : FinMeasAdditive μ T) (hT' : FinMeasAdditive μ T') :
FinMeasAdditive μ (T + T') := by
intro s t hs ht hμs hμt hst
simp only [hT s t hs ht hμs hμt hst, hT' s t hs ht hμs hμt hst, Pi.add_apply]
abel
#align measure_theory.fin_meas_additive.add MeasureTheory.FinMeasAdditive.add
theorem smul [Monoid 𝕜] [DistribMulAction 𝕜 β] (hT : FinMeasAdditive μ T) (c : 𝕜) :
FinMeasAdditive μ fun s => c • T s := fun s t hs ht hμs hμt hst => by
simp [hT s t hs ht hμs hμt hst]
#align measure_theory.fin_meas_additive.smul MeasureTheory.FinMeasAdditive.smul
theorem of_eq_top_imp_eq_top {μ' : Measure α} (h : ∀ s, MeasurableSet s → μ s = ∞ → μ' s = ∞)
(hT : FinMeasAdditive μ T) : FinMeasAdditive μ' T := fun s t hs ht hμ's hμ't hst =>
hT s t hs ht (mt (h s hs) hμ's) (mt (h t ht) hμ't) hst
#align measure_theory.fin_meas_additive.of_eq_top_imp_eq_top MeasureTheory.FinMeasAdditive.of_eq_top_imp_eq_top
theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
#align measure_theory.fin_meas_additive.of_smul_measure MeasureTheory.FinMeasAdditive.of_smul_measure
theorem smul_measure (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hT : FinMeasAdditive μ T) :
FinMeasAdditive (c • μ) T := by
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top]
simp only [hc_ne_zero, true_and_iff, Ne, not_false_iff]
exact Or.inl hμs
#align measure_theory.fin_meas_additive.smul_measure MeasureTheory.FinMeasAdditive.smul_measure
theorem smul_measure_iff (c : ℝ≥0∞) (hc_ne_zero : c ≠ 0) (hc_ne_top : c ≠ ∞) :
FinMeasAdditive (c • μ) T ↔ FinMeasAdditive μ T :=
⟨fun hT => of_smul_measure c hc_ne_top hT, fun hT => smul_measure c hc_ne_zero hT⟩
#align measure_theory.fin_meas_additive.smul_measure_iff MeasureTheory.FinMeasAdditive.smul_measure_iff
theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) :
T ∅ = 0 := by
have h_empty : μ ∅ ≠ ∞ := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne
specialize hT ∅ ∅ MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty ∅)
rw [Set.union_empty] at hT
nth_rw 1 [← add_zero (T ∅)] at hT
exact (add_left_cancel hT).symm
#align measure_theory.fin_meas_additive.map_empty_eq_zero MeasureTheory.FinMeasAdditive.map_empty_eq_zero
| Mathlib/MeasureTheory/Integral/SetToL1.lean | 152 | 179 | theorem map_iUnion_fin_meas_set_eq_sum (T : Set α → β) (T_empty : T ∅ = 0)
(h_add : FinMeasAdditive μ T) {ι} (S : ι → Set α) (sι : Finset ι)
(hS_meas : ∀ i, MeasurableSet (S i)) (hSp : ∀ i ∈ sι, μ (S i) ≠ ∞)
(h_disj : ∀ᵉ (i ∈ sι) (j ∈ sι), i ≠ j → Disjoint (S i) (S j)) :
T (⋃ i ∈ sι, S i) = ∑ i ∈ sι, T (S i) := by |
revert hSp h_disj
refine Finset.induction_on sι ?_ ?_
· simp only [Finset.not_mem_empty, IsEmpty.forall_iff, iUnion_false, iUnion_empty, sum_empty,
forall₂_true_iff, imp_true_iff, forall_true_left, not_false_iff, T_empty]
intro a s has h hps h_disj
rw [Finset.sum_insert has, ← h]
swap; · exact fun i hi => hps i (Finset.mem_insert_of_mem hi)
swap;
· exact fun i hi j hj hij =>
h_disj i (Finset.mem_insert_of_mem hi) j (Finset.mem_insert_of_mem hj) hij
rw [←
h_add (S a) (⋃ i ∈ s, S i) (hS_meas a) (measurableSet_biUnion _ fun i _ => hS_meas i)
(hps a (Finset.mem_insert_self a s))]
· congr; convert Finset.iSup_insert a s S
· exact
((measure_biUnion_finset_le _ _).trans_lt <|
ENNReal.sum_lt_top fun i hi => hps i <| Finset.mem_insert_of_mem hi).ne
· simp_rw [Set.inter_iUnion]
refine iUnion_eq_empty.mpr fun i => iUnion_eq_empty.mpr fun hi => ?_
rw [← Set.disjoint_iff_inter_eq_empty]
refine h_disj a (Finset.mem_insert_self a s) i (Finset.mem_insert_of_mem hi) fun hai => ?_
rw [← hai] at hi
exact has hi
| 2,117 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
| Mathlib/MeasureTheory/Integral/Bochner.lean | 171 | 172 | theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by | simp [weightedSMul]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
| Mathlib/MeasureTheory/Integral/Bochner.lean | 176 | 177 | theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by | ext1; simp [weightedSMul]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
| Mathlib/MeasureTheory/Integral/Bochner.lean | 181 | 182 | theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by | ext1 x; rw [weightedSMul_apply]; simp
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
| Mathlib/MeasureTheory/Integral/Bochner.lean | 185 | 192 | theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by |
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
| Mathlib/MeasureTheory/Integral/Bochner.lean | 195 | 201 | theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by |
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
#align measure_theory.weighted_smul_smul_measure MeasureTheory.weightedSMul_smul_measure
| Mathlib/MeasureTheory/Integral/Bochner.lean | 204 | 206 | theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by |
ext1 x; simp_rw [weightedSMul_apply]; congr 2
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
#align measure_theory.weighted_smul_smul_measure MeasureTheory.weightedSMul_smul_measure
theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by
ext1 x; simp_rw [weightedSMul_apply]; congr 2
#align measure_theory.weighted_smul_congr MeasureTheory.weightedSMul_congr
| Mathlib/MeasureTheory/Integral/Bochner.lean | 209 | 210 | theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 := by |
ext1 x; rw [weightedSMul_apply, h_zero]; simp
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
#align measure_theory.weighted_smul_smul_measure MeasureTheory.weightedSMul_smul_measure
theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by
ext1 x; simp_rw [weightedSMul_apply]; congr 2
#align measure_theory.weighted_smul_congr MeasureTheory.weightedSMul_congr
theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 := by
ext1 x; rw [weightedSMul_apply, h_zero]; simp
#align measure_theory.weighted_smul_null MeasureTheory.weightedSMul_null
| Mathlib/MeasureTheory/Integral/Bochner.lean | 213 | 219 | theorem weightedSMul_union' (s t : Set α) (ht : MeasurableSet t) (hs_finite : μ s ≠ ∞)
(ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t := by |
ext1 x
simp_rw [add_apply, weightedSMul_apply,
measure_union (Set.disjoint_iff_inter_eq_empty.mpr h_inter) ht,
ENNReal.toReal_add hs_finite ht_finite, add_smul]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
#align measure_theory.weighted_smul_smul_measure MeasureTheory.weightedSMul_smul_measure
theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by
ext1 x; simp_rw [weightedSMul_apply]; congr 2
#align measure_theory.weighted_smul_congr MeasureTheory.weightedSMul_congr
theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 := by
ext1 x; rw [weightedSMul_apply, h_zero]; simp
#align measure_theory.weighted_smul_null MeasureTheory.weightedSMul_null
theorem weightedSMul_union' (s t : Set α) (ht : MeasurableSet t) (hs_finite : μ s ≠ ∞)
(ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t := by
ext1 x
simp_rw [add_apply, weightedSMul_apply,
measure_union (Set.disjoint_iff_inter_eq_empty.mpr h_inter) ht,
ENNReal.toReal_add hs_finite ht_finite, add_smul]
#align measure_theory.weighted_smul_union' MeasureTheory.weightedSMul_union'
@[nolint unusedArguments]
theorem weightedSMul_union (s t : Set α) (_hs : MeasurableSet s) (ht : MeasurableSet t)
(hs_finite : μ s ≠ ∞) (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t :=
weightedSMul_union' s t ht hs_finite ht_finite h_inter
#align measure_theory.weighted_smul_union MeasureTheory.weightedSMul_union
| Mathlib/MeasureTheory/Integral/Bochner.lean | 229 | 231 | theorem weightedSMul_smul [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜)
(s : Set α) (x : F) : weightedSMul μ s (c • x) = c • weightedSMul μ s x := by |
simp_rw [weightedSMul_apply, smul_comm]
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
section WeightedSMul
open ContinuousLinearMap
variable [NormedAddCommGroup F] [NormedSpace ℝ F] {m : MeasurableSpace α} {μ : Measure α}
def weightedSMul {_ : MeasurableSpace α} (μ : Measure α) (s : Set α) : F →L[ℝ] F :=
(μ s).toReal • ContinuousLinearMap.id ℝ F
#align measure_theory.weighted_smul MeasureTheory.weightedSMul
theorem weightedSMul_apply {m : MeasurableSpace α} (μ : Measure α) (s : Set α) (x : F) :
weightedSMul μ s x = (μ s).toReal • x := by simp [weightedSMul]
#align measure_theory.weighted_smul_apply MeasureTheory.weightedSMul_apply
@[simp]
theorem weightedSMul_zero_measure {m : MeasurableSpace α} :
weightedSMul (0 : Measure α) = (0 : Set α → F →L[ℝ] F) := by ext1; simp [weightedSMul]
#align measure_theory.weighted_smul_zero_measure MeasureTheory.weightedSMul_zero_measure
@[simp]
theorem weightedSMul_empty {m : MeasurableSpace α} (μ : Measure α) :
weightedSMul μ ∅ = (0 : F →L[ℝ] F) := by ext1 x; rw [weightedSMul_apply]; simp
#align measure_theory.weighted_smul_empty MeasureTheory.weightedSMul_empty
theorem weightedSMul_add_measure {m : MeasurableSpace α} (μ ν : Measure α) {s : Set α}
(hμs : μ s ≠ ∞) (hνs : ν s ≠ ∞) :
(weightedSMul (μ + ν) s : F →L[ℝ] F) = weightedSMul μ s + weightedSMul ν s := by
ext1 x
push_cast
simp_rw [Pi.add_apply, weightedSMul_apply]
push_cast
rw [Pi.add_apply, ENNReal.toReal_add hμs hνs, add_smul]
#align measure_theory.weighted_smul_add_measure MeasureTheory.weightedSMul_add_measure
theorem weightedSMul_smul_measure {m : MeasurableSpace α} (μ : Measure α) (c : ℝ≥0∞) {s : Set α} :
(weightedSMul (c • μ) s : F →L[ℝ] F) = c.toReal • weightedSMul μ s := by
ext1 x
push_cast
simp_rw [Pi.smul_apply, weightedSMul_apply]
push_cast
simp_rw [Pi.smul_apply, smul_eq_mul, toReal_mul, smul_smul]
#align measure_theory.weighted_smul_smul_measure MeasureTheory.weightedSMul_smul_measure
theorem weightedSMul_congr (s t : Set α) (hst : μ s = μ t) :
(weightedSMul μ s : F →L[ℝ] F) = weightedSMul μ t := by
ext1 x; simp_rw [weightedSMul_apply]; congr 2
#align measure_theory.weighted_smul_congr MeasureTheory.weightedSMul_congr
theorem weightedSMul_null {s : Set α} (h_zero : μ s = 0) : (weightedSMul μ s : F →L[ℝ] F) = 0 := by
ext1 x; rw [weightedSMul_apply, h_zero]; simp
#align measure_theory.weighted_smul_null MeasureTheory.weightedSMul_null
theorem weightedSMul_union' (s t : Set α) (ht : MeasurableSet t) (hs_finite : μ s ≠ ∞)
(ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t := by
ext1 x
simp_rw [add_apply, weightedSMul_apply,
measure_union (Set.disjoint_iff_inter_eq_empty.mpr h_inter) ht,
ENNReal.toReal_add hs_finite ht_finite, add_smul]
#align measure_theory.weighted_smul_union' MeasureTheory.weightedSMul_union'
@[nolint unusedArguments]
theorem weightedSMul_union (s t : Set α) (_hs : MeasurableSet s) (ht : MeasurableSet t)
(hs_finite : μ s ≠ ∞) (ht_finite : μ t ≠ ∞) (h_inter : s ∩ t = ∅) :
(weightedSMul μ (s ∪ t) : F →L[ℝ] F) = weightedSMul μ s + weightedSMul μ t :=
weightedSMul_union' s t ht hs_finite ht_finite h_inter
#align measure_theory.weighted_smul_union MeasureTheory.weightedSMul_union
theorem weightedSMul_smul [NormedField 𝕜] [NormedSpace 𝕜 F] [SMulCommClass ℝ 𝕜 F] (c : 𝕜)
(s : Set α) (x : F) : weightedSMul μ s (c • x) = c • weightedSMul μ s x := by
simp_rw [weightedSMul_apply, smul_comm]
#align measure_theory.weighted_smul_smul MeasureTheory.weightedSMul_smul
theorem norm_weightedSMul_le (s : Set α) : ‖(weightedSMul μ s : F →L[ℝ] F)‖ ≤ (μ s).toReal :=
calc
‖(weightedSMul μ s : F →L[ℝ] F)‖ = ‖(μ s).toReal‖ * ‖ContinuousLinearMap.id ℝ F‖ :=
norm_smul (μ s).toReal (ContinuousLinearMap.id ℝ F)
_ ≤ ‖(μ s).toReal‖ :=
((mul_le_mul_of_nonneg_left norm_id_le (norm_nonneg _)).trans (mul_one _).le)
_ = abs (μ s).toReal := Real.norm_eq_abs _
_ = (μ s).toReal := abs_eq_self.mpr ENNReal.toReal_nonneg
#align measure_theory.norm_weighted_smul_le MeasureTheory.norm_weightedSMul_le
theorem dominatedFinMeasAdditive_weightedSMul {_ : MeasurableSpace α} (μ : Measure α) :
DominatedFinMeasAdditive μ (weightedSMul μ : Set α → F →L[ℝ] F) 1 :=
⟨weightedSMul_union, fun s _ _ => (norm_weightedSMul_le s).trans (one_mul _).symm.le⟩
#align measure_theory.dominated_fin_meas_additive_weighted_smul MeasureTheory.dominatedFinMeasAdditive_weightedSMul
| Mathlib/MeasureTheory/Integral/Bochner.lean | 249 | 251 | theorem weightedSMul_nonneg (s : Set α) (x : ℝ) (hx : 0 ≤ x) : 0 ≤ weightedSMul μ s x := by |
simp only [weightedSMul, Algebra.id.smul_eq_mul, coe_smul', _root_.id, coe_id', Pi.smul_apply]
exact mul_nonneg toReal_nonneg hx
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
local infixr:25 " →ₛ " => SimpleFunc
namespace SimpleFunc
section PosPart
variable [LinearOrder E] [Zero E] [MeasurableSpace α]
def posPart (f : α →ₛ E) : α →ₛ E :=
f.map fun b => max b 0
#align measure_theory.simple_func.pos_part MeasureTheory.SimpleFunc.posPart
def negPart [Neg E] (f : α →ₛ E) : α →ₛ E :=
posPart (-f)
#align measure_theory.simple_func.neg_part MeasureTheory.SimpleFunc.negPart
| Mathlib/MeasureTheory/Integral/Bochner.lean | 274 | 275 | theorem posPart_map_norm (f : α →ₛ ℝ) : (posPart f).map norm = posPart f := by |
ext; rw [map_apply, Real.norm_eq_abs, abs_of_nonneg]; exact le_max_right _ _
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
local infixr:25 " →ₛ " => SimpleFunc
namespace SimpleFunc
section PosPart
variable [LinearOrder E] [Zero E] [MeasurableSpace α]
def posPart (f : α →ₛ E) : α →ₛ E :=
f.map fun b => max b 0
#align measure_theory.simple_func.pos_part MeasureTheory.SimpleFunc.posPart
def negPart [Neg E] (f : α →ₛ E) : α →ₛ E :=
posPart (-f)
#align measure_theory.simple_func.neg_part MeasureTheory.SimpleFunc.negPart
theorem posPart_map_norm (f : α →ₛ ℝ) : (posPart f).map norm = posPart f := by
ext; rw [map_apply, Real.norm_eq_abs, abs_of_nonneg]; exact le_max_right _ _
#align measure_theory.simple_func.pos_part_map_norm MeasureTheory.SimpleFunc.posPart_map_norm
| Mathlib/MeasureTheory/Integral/Bochner.lean | 278 | 279 | theorem negPart_map_norm (f : α →ₛ ℝ) : (negPart f).map norm = negPart f := by |
rw [negPart]; exact posPart_map_norm _
| 2,118 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace ENNReal EMetric
namespace MeasureTheory
variable {α E F 𝕜 : Type*}
local infixr:25 " →ₛ " => SimpleFunc
namespace SimpleFunc
section PosPart
variable [LinearOrder E] [Zero E] [MeasurableSpace α]
def posPart (f : α →ₛ E) : α →ₛ E :=
f.map fun b => max b 0
#align measure_theory.simple_func.pos_part MeasureTheory.SimpleFunc.posPart
def negPart [Neg E] (f : α →ₛ E) : α →ₛ E :=
posPart (-f)
#align measure_theory.simple_func.neg_part MeasureTheory.SimpleFunc.negPart
theorem posPart_map_norm (f : α →ₛ ℝ) : (posPart f).map norm = posPart f := by
ext; rw [map_apply, Real.norm_eq_abs, abs_of_nonneg]; exact le_max_right _ _
#align measure_theory.simple_func.pos_part_map_norm MeasureTheory.SimpleFunc.posPart_map_norm
theorem negPart_map_norm (f : α →ₛ ℝ) : (negPart f).map norm = negPart f := by
rw [negPart]; exact posPart_map_norm _
#align measure_theory.simple_func.neg_part_map_norm MeasureTheory.SimpleFunc.negPart_map_norm
| Mathlib/MeasureTheory/Integral/Bochner.lean | 282 | 286 | theorem posPart_sub_negPart (f : α →ₛ ℝ) : f.posPart - f.negPart = f := by |
simp only [posPart, negPart]
ext a
rw [coe_sub]
exact max_zero_sub_eq_self (f a)
| 2,118 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableInv
variable [Group G] [MeasurableInv G]
@[to_additive]
theorem Integrable.comp_inv [IsInvInvariant μ] {f : G → F} (hf : Integrable f μ) :
Integrable (fun t => f t⁻¹) μ :=
(hf.mono_measure (map_inv_eq_self μ).le).comp_measurable measurable_inv
#align measure_theory.integrable.comp_inv MeasureTheory.Integrable.comp_inv
#align measure_theory.integrable.comp_neg MeasureTheory.Integrable.comp_neg
@[to_additive]
| Mathlib/MeasureTheory/Group/Integral.lean | 40 | 43 | theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
| Mathlib/MeasureTheory/Group/Integral.lean | 58 | 61 | theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by |
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
#align measure_theory.integral_mul_left_eq_self MeasureTheory.integral_mul_left_eq_self
#align measure_theory.integral_add_left_eq_self MeasureTheory.integral_add_left_eq_self
@[to_additive
"Translating a function by right-addition does not change its integral with respect to a
right-invariant measure."] -- Porting note: was `@[simp]`
| Mathlib/MeasureTheory/Group/Integral.lean | 70 | 74 | theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by |
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
#align measure_theory.integral_mul_left_eq_self MeasureTheory.integral_mul_left_eq_self
#align measure_theory.integral_add_left_eq_self MeasureTheory.integral_add_left_eq_self
@[to_additive
"Translating a function by right-addition does not change its integral with respect to a
right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
#align measure_theory.integral_mul_right_eq_self MeasureTheory.integral_mul_right_eq_self
#align measure_theory.integral_add_right_eq_self MeasureTheory.integral_add_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
| Mathlib/MeasureTheory/Group/Integral.lean | 79 | 83 | theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by |
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
#align measure_theory.integral_mul_left_eq_self MeasureTheory.integral_mul_left_eq_self
#align measure_theory.integral_add_left_eq_self MeasureTheory.integral_add_left_eq_self
@[to_additive
"Translating a function by right-addition does not change its integral with respect to a
right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
#align measure_theory.integral_mul_right_eq_self MeasureTheory.integral_mul_right_eq_self
#align measure_theory.integral_add_right_eq_self MeasureTheory.integral_add_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
#align measure_theory.integral_div_right_eq_self MeasureTheory.integral_div_right_eq_self
#align measure_theory.integral_sub_right_eq_self MeasureTheory.integral_sub_right_eq_self
@[to_additive
"If some left-translate of a function negates it, then the integral of the function with
respect to a left-invariant measure is 0."]
| Mathlib/MeasureTheory/Group/Integral.lean | 92 | 94 | theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
∫ x, f x ∂μ = 0 := by |
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
#align measure_theory.integral_mul_left_eq_self MeasureTheory.integral_mul_left_eq_self
#align measure_theory.integral_add_left_eq_self MeasureTheory.integral_add_left_eq_self
@[to_additive
"Translating a function by right-addition does not change its integral with respect to a
right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
#align measure_theory.integral_mul_right_eq_self MeasureTheory.integral_mul_right_eq_self
#align measure_theory.integral_add_right_eq_self MeasureTheory.integral_add_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
#align measure_theory.integral_div_right_eq_self MeasureTheory.integral_div_right_eq_self
#align measure_theory.integral_sub_right_eq_self MeasureTheory.integral_sub_right_eq_self
@[to_additive
"If some left-translate of a function negates it, then the integral of the function with
respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
∫ x, f x ∂μ = 0 := by
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self]
#align measure_theory.integral_eq_zero_of_mul_left_eq_neg MeasureTheory.integral_eq_zero_of_mul_left_eq_neg
#align measure_theory.integral_eq_zero_of_add_left_eq_neg MeasureTheory.integral_eq_zero_of_add_left_eq_neg
@[to_additive
"If some right-translate of a function negates it, then the integral of the function with
respect to a right-invariant measure is 0."]
| Mathlib/MeasureTheory/Group/Integral.lean | 103 | 105 | theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
∫ x, f x ∂μ = 0 := by |
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self]
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a function by left-addition does not change its integral with respect to a
left-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → E) (g : G) :
(∫ x, f (g * x) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => g * x := (MeasurableEquiv.mulLeft g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_left_eq_self]
#align measure_theory.integral_mul_left_eq_self MeasureTheory.integral_mul_left_eq_self
#align measure_theory.integral_add_left_eq_self MeasureTheory.integral_add_left_eq_self
@[to_additive
"Translating a function by right-addition does not change its integral with respect to a
right-invariant measure."] -- Porting note: was `@[simp]`
theorem integral_mul_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x * g) ∂μ) = ∫ x, f x ∂μ := by
have h_mul : MeasurableEmbedding fun x => x * g :=
(MeasurableEquiv.mulRight g).measurableEmbedding
rw [← h_mul.integral_map, map_mul_right_eq_self]
#align measure_theory.integral_mul_right_eq_self MeasureTheory.integral_mul_right_eq_self
#align measure_theory.integral_add_right_eq_self MeasureTheory.integral_add_right_eq_self
@[to_additive] -- Porting note: was `@[simp]`
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by
simp_rw [div_eq_mul_inv]
-- Porting note: was `simp_rw`
rw [integral_mul_right_eq_self f g⁻¹]
#align measure_theory.integral_div_right_eq_self MeasureTheory.integral_div_right_eq_self
#align measure_theory.integral_sub_right_eq_self MeasureTheory.integral_sub_right_eq_self
@[to_additive
"If some left-translate of a function negates it, then the integral of the function with
respect to a left-invariant measure is 0."]
theorem integral_eq_zero_of_mul_left_eq_neg [IsMulLeftInvariant μ] (hf' : ∀ x, f (g * x) = -f x) :
∫ x, f x ∂μ = 0 := by
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_left_eq_self]
#align measure_theory.integral_eq_zero_of_mul_left_eq_neg MeasureTheory.integral_eq_zero_of_mul_left_eq_neg
#align measure_theory.integral_eq_zero_of_add_left_eq_neg MeasureTheory.integral_eq_zero_of_add_left_eq_neg
@[to_additive
"If some right-translate of a function negates it, then the integral of the function with
respect to a right-invariant measure is 0."]
theorem integral_eq_zero_of_mul_right_eq_neg [IsMulRightInvariant μ] (hf' : ∀ x, f (x * g) = -f x) :
∫ x, f x ∂μ = 0 := by
simp_rw [← self_eq_neg ℝ E, ← integral_neg, ← hf', integral_mul_right_eq_self]
#align measure_theory.integral_eq_zero_of_mul_right_eq_neg MeasureTheory.integral_eq_zero_of_mul_right_eq_neg
#align measure_theory.integral_eq_zero_of_add_right_eq_neg MeasureTheory.integral_eq_zero_of_add_right_eq_neg
@[to_additive]
theorem Integrable.comp_mul_left {f : G → F} [IsMulLeftInvariant μ] (hf : Integrable f μ) (g : G) :
Integrable (fun t => f (g * t)) μ :=
(hf.mono_measure (map_mul_left_eq_self μ g).le).comp_measurable <| measurable_const_mul g
#align measure_theory.integrable.comp_mul_left MeasureTheory.Integrable.comp_mul_left
#align measure_theory.integrable.comp_add_left MeasureTheory.Integrable.comp_add_left
@[to_additive]
theorem Integrable.comp_mul_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
(g : G) : Integrable (fun t => f (t * g)) μ :=
(hf.mono_measure (map_mul_right_eq_self μ g).le).comp_measurable <| measurable_mul_const g
#align measure_theory.integrable.comp_mul_right MeasureTheory.Integrable.comp_mul_right
#align measure_theory.integrable.comp_add_right MeasureTheory.Integrable.comp_add_right
@[to_additive]
| Mathlib/MeasureTheory/Group/Integral.lean | 124 | 127 | theorem Integrable.comp_div_right {f : G → F} [IsMulRightInvariant μ] (hf : Integrable f μ)
(g : G) : Integrable (fun t => f (t / g)) μ := by |
simp_rw [div_eq_mul_inv]
exact hf.comp_mul_right g⁻¹
| 2,119 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F : Type*} [MeasurableSpace G]
variable [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] [NormedAddCommGroup F]
variable {μ : Measure G} {f : G → E} {g : G}
section SMul
variable [Group G] [MeasurableSpace α] [MulAction G α] [MeasurableSMul G α]
@[to_additive] -- Porting note: was `@[simp]`
| Mathlib/MeasureTheory/Group/Integral.lean | 165 | 168 | theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
(∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : α => g • x := (MeasurableEquiv.smul g).measurableEmbedding
rw [← h.integral_map, map_smul]
| 2,119 |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open scoped ENNReal NNReal
open MeasureTheory MeasureTheory.Measure
variable {α : Type*} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : Measure α)
[WeaklyRegular μ]
namespace MeasureTheory
local infixr:25 " →ₛ " => SimpleFunc
| Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 93 | 152 | theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
(ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧
(∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε := by |
induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
· let f := SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)
by_cases h : ∫⁻ x, f x ∂μ = ⊤
· refine
⟨fun _ => c, fun x => ?_, lowerSemicontinuous_const, by
simp only [_root_.top_add, le_top, h]⟩
simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
exact Set.indicator_le_self _ _ _
by_cases hc : c = 0
· refine ⟨fun _ => 0, ?_, lowerSemicontinuous_const, ?_⟩
· classical
simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
SimpleFunc.coe_piecewise, le_zero_iff]
· simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
have ne_top : μ s ≠ ⊤ := by
classical
simpa [f, hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,
restrict_apply] using h
have : μ s < μ s + ε / c := by
have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
simpa using ENNReal.add_lt_add_left ne_top this
obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _), u ⊇ s ∧ IsOpen u ∧ μ u < μ s + ε / c :=
s.exists_isOpen_lt_of_lt _ this
refine ⟨Set.indicator u fun _ => c,
fun x => ?_, u_open.lowerSemicontinuous_indicator (zero_le _), ?_⟩
· simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
exact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _
· suffices (c : ℝ≥0∞) * μ u ≤ c * μ s + ε by
classical
simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,
lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,
coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]
calc
(c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := mul_le_mul_left' μu.le _
_ = c * μ s + ε := by
simp_rw [mul_add]
rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
simpa using hc
· rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
refine
⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, ?_⟩
simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]
rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
convert add_le_add g₁int g₂int using 1
conv_lhs => rw [← ENNReal.add_halves ε]
abel
| 2,120 |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open scoped ENNReal NNReal
open MeasureTheory MeasureTheory.Measure
variable {α : Type*} [TopologicalSpace α] [MeasurableSpace α] [BorelSpace α] (μ : Measure α)
[WeaklyRegular μ]
namespace MeasureTheory
local infixr:25 " →ₛ " => SimpleFunc
theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : α →ₛ ℝ≥0) {ε : ℝ≥0∞}
(ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0, (∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧
(∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε := by
induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
· let f := SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)
by_cases h : ∫⁻ x, f x ∂μ = ⊤
· refine
⟨fun _ => c, fun x => ?_, lowerSemicontinuous_const, by
simp only [_root_.top_add, le_top, h]⟩
simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
exact Set.indicator_le_self _ _ _
by_cases hc : c = 0
· refine ⟨fun _ => 0, ?_, lowerSemicontinuous_const, ?_⟩
· classical
simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,
SimpleFunc.coe_piecewise, le_zero_iff]
· simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero]
have ne_top : μ s ≠ ⊤ := by
classical
simpa [f, hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const,
Function.const_apply, lintegral_const, ENNReal.coe_indicator, Set.univ_inter,
ENNReal.coe_ne_top, MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,
or_false_iff, lintegral_indicator, ENNReal.coe_eq_zero, Ne, not_false_iff,
SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, false_and_iff,
restrict_apply] using h
have : μ s < μ s + ε / c := by
have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩
simpa using ENNReal.add_lt_add_left ne_top this
obtain ⟨u, su, u_open, μu⟩ : ∃ (u : _), u ⊇ s ∧ IsOpen u ∧ μ u < μ s + ε / c :=
s.exists_isOpen_lt_of_lt _ this
refine ⟨Set.indicator u fun _ => c,
fun x => ?_, u_open.lowerSemicontinuous_indicator (zero_le _), ?_⟩
· simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,
Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]
exact Set.indicator_le_indicator_of_subset su (fun x => zero_le _) _
· suffices (c : ℝ≥0∞) * μ u ≤ c * μ s + ε by
classical
simpa only [ENNReal.coe_indicator, u_open.measurableSet, lintegral_indicator,
lintegral_const, MeasurableSet.univ, Measure.restrict_apply, Set.univ_inter, const_zero,
coe_piecewise, coe_const, coe_zero, Set.piecewise_eq_indicator, Function.const_apply, hs]
calc
(c : ℝ≥0∞) * μ u ≤ c * (μ s + ε / c) := mul_le_mul_left' μu.le _
_ = c * μ s + ε := by
simp_rw [mul_add]
rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]
simpa using hc
· rcases h₁ (ENNReal.half_pos ε0).ne' with ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩
rcases h₂ (ENNReal.half_pos ε0).ne' with ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩
refine
⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, ?_⟩
simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]
rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,
lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]
convert add_le_add g₁int g₂int using 1
conv_lhs => rw [← ENNReal.add_halves ε]
abel
#align measure_theory.simple_func.exists_le_lower_semicontinuous_lintegral_ge MeasureTheory.SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge
-- Porting note: errors with
-- `ambiguous identifier 'eapproxDiff', possible interpretations:`
-- `[SimpleFunc.eapproxDiff, SimpleFunc.eapproxDiff]`
-- open SimpleFunc (eapproxDiff tsum_eapproxDiff)
| Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 164 | 195 | theorem exists_le_lowerSemicontinuous_lintegral_ge (f : α → ℝ≥0∞) (hf : Measurable f) {ε : ℝ≥0∞}
(εpos : ε ≠ 0) :
∃ g : α → ℝ≥0∞,
(∀ x, f x ≤ g x) ∧ LowerSemicontinuous g ∧ (∫⁻ x, g x ∂μ) ≤ (∫⁻ x, f x ∂μ) + ε := by |
rcases ENNReal.exists_pos_sum_of_countable' εpos ℕ with ⟨δ, δpos, hδ⟩
have :
∀ n,
∃ g : α → ℝ≥0,
(∀ x, SimpleFunc.eapproxDiff f n x ≤ g x) ∧
LowerSemicontinuous g ∧
(∫⁻ x, g x ∂μ) ≤ (∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n :=
fun n =>
SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge μ (SimpleFunc.eapproxDiff f n)
(δpos n).ne'
choose g f_le_g gcont hg using this
refine ⟨fun x => ∑' n, g n x, fun x => ?_, ?_, ?_⟩
· rw [← SimpleFunc.tsum_eapproxDiff f hf]
exact ENNReal.tsum_le_tsum fun n => ENNReal.coe_le_coe.2 (f_le_g n x)
· refine lowerSemicontinuous_tsum fun n => ?_
exact
ENNReal.continuous_coe.comp_lowerSemicontinuous (gcont n) fun x y hxy =>
ENNReal.coe_le_coe.2 hxy
· calc
∫⁻ x, ∑' n : ℕ, g n x ∂μ = ∑' n, ∫⁻ x, g n x ∂μ := by
rw [lintegral_tsum fun n => (gcont n).measurable.coe_nnreal_ennreal.aemeasurable]
_ ≤ ∑' n, ((∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ) + δ n) := ENNReal.tsum_le_tsum hg
_ = ∑' n, ∫⁻ x, SimpleFunc.eapproxDiff f n x ∂μ + ∑' n, δ n := ENNReal.tsum_add
_ ≤ (∫⁻ x : α, f x ∂μ) + ε := by
refine add_le_add ?_ hδ.le
rw [← lintegral_tsum]
· simp_rw [SimpleFunc.tsum_eapproxDiff f hf, le_refl]
· intro n; exact (SimpleFunc.measurable _).coe_nnreal_ennreal.aemeasurable
| 2,120 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 62 | 64 | theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by |
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
#align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
AEStronglyMeasurable' m' f μ :=
let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 71 | 75 | theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
(hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by |
rcases hf with ⟨f', h_f'_meas, hff'⟩
rcases hg with ⟨g', h_g'_meas, hgg'⟩
exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
#align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
AEStronglyMeasurable' m' f μ :=
let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩
theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
(hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by
rcases hf with ⟨f', h_f'_meas, hff'⟩
rcases hg with ⟨g', h_g'_meas, hgg'⟩
exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩
#align measure_theory.ae_strongly_measurable'.add MeasureTheory.AEStronglyMeasurable'.add
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 78 | 83 | theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
#align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
AEStronglyMeasurable' m' f μ :=
let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩
theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
(hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by
rcases hf with ⟨f', h_f'_meas, hff'⟩
rcases hg with ⟨g', h_g'_meas, hgg'⟩
exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩
#align measure_theory.ae_strongly_measurable'.add MeasureTheory.AEStronglyMeasurable'.add
theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
#align measure_theory.ae_strongly_measurable'.neg MeasureTheory.AEStronglyMeasurable'.neg
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 86 | 92 | theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
(hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
rcases hgm with ⟨g', hg'_meas, hg_ae⟩
refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
simp_rw [Pi.sub_apply]
rw [hx1, hx2]
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
#align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
AEStronglyMeasurable' m' f μ :=
let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩
theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
(hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by
rcases hf with ⟨f', h_f'_meas, hff'⟩
rcases hg with ⟨g', h_g'_meas, hgg'⟩
exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩
#align measure_theory.ae_strongly_measurable'.add MeasureTheory.AEStronglyMeasurable'.add
theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
#align measure_theory.ae_strongly_measurable'.neg MeasureTheory.AEStronglyMeasurable'.neg
theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
(hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
rcases hgm with ⟨g', hg'_meas, hg_ae⟩
refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
simp_rw [Pi.sub_apply]
rw [hx1, hx2]
#align measure_theory.ae_strongly_measurable'.sub MeasureTheory.AEStronglyMeasurable'.sub
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 95 | 99 | theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (c • f) μ := by |
rcases hf with ⟨f', h_f'_meas, hff'⟩
refine ⟨c • f', h_f'_meas.const_smul c, ?_⟩
exact EventuallyEq.fun_comp hff' fun x => c • x
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
namespace AEStronglyMeasurable'
variable {α β 𝕜 : Type*} {m m0 : MeasurableSpace α} {μ : Measure α} [TopologicalSpace β]
{f g : α → β}
theorem congr (hf : AEStronglyMeasurable' m f μ) (hfg : f =ᵐ[μ] g) :
AEStronglyMeasurable' m g μ := by
obtain ⟨f', hf'_meas, hff'⟩ := hf; exact ⟨f', hf'_meas, hfg.symm.trans hff'⟩
#align measure_theory.ae_strongly_measurable'.congr MeasureTheory.AEStronglyMeasurable'.congr
theorem mono {m'} (hf : AEStronglyMeasurable' m f μ) (hm : m ≤ m') :
AEStronglyMeasurable' m' f μ :=
let ⟨f', hf'_meas, hff'⟩ := hf; ⟨f', hf'_meas.mono hm, hff'⟩
theorem add [Add β] [ContinuousAdd β] (hf : AEStronglyMeasurable' m f μ)
(hg : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f + g) μ := by
rcases hf with ⟨f', h_f'_meas, hff'⟩
rcases hg with ⟨g', h_g'_meas, hgg'⟩
exact ⟨f' + g', h_f'_meas.add h_g'_meas, hff'.add hgg'⟩
#align measure_theory.ae_strongly_measurable'.add MeasureTheory.AEStronglyMeasurable'.add
theorem neg [AddGroup β] [TopologicalAddGroup β] {f : α → β} (hfm : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (-f) μ := by
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine ⟨-f', hf'_meas.neg, hf_ae.mono fun x hx => ?_⟩
simp_rw [Pi.neg_apply]
rw [hx]
#align measure_theory.ae_strongly_measurable'.neg MeasureTheory.AEStronglyMeasurable'.neg
theorem sub [AddGroup β] [TopologicalAddGroup β] {f g : α → β} (hfm : AEStronglyMeasurable' m f μ)
(hgm : AEStronglyMeasurable' m g μ) : AEStronglyMeasurable' m (f - g) μ := by
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
rcases hgm with ⟨g', hg'_meas, hg_ae⟩
refine ⟨f' - g', hf'_meas.sub hg'_meas, hf_ae.mp (hg_ae.mono fun x hx1 hx2 => ?_)⟩
simp_rw [Pi.sub_apply]
rw [hx1, hx2]
#align measure_theory.ae_strongly_measurable'.sub MeasureTheory.AEStronglyMeasurable'.sub
theorem const_smul [SMul 𝕜 β] [ContinuousConstSMul 𝕜 β] (c : 𝕜) (hf : AEStronglyMeasurable' m f μ) :
AEStronglyMeasurable' m (c • f) μ := by
rcases hf with ⟨f', h_f'_meas, hff'⟩
refine ⟨c • f', h_f'_meas.const_smul c, ?_⟩
exact EventuallyEq.fun_comp hff' fun x => c • x
#align measure_theory.ae_strongly_measurable'.const_smul MeasureTheory.AEStronglyMeasurable'.const_smul
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 102 | 110 | theorem const_inner {𝕜 β} [RCLike 𝕜] [NormedAddCommGroup β] [InnerProductSpace 𝕜 β] {f : α → β}
(hfm : AEStronglyMeasurable' m f μ) (c : β) :
AEStronglyMeasurable' m (fun x => (inner c (f x) : 𝕜)) μ := by |
rcases hfm with ⟨f', hf'_meas, hf_ae⟩
refine
⟨fun x => (inner c (f' x) : 𝕜), (@stronglyMeasurable_const _ _ m _ c).inner hf'_meas,
hf_ae.mono fun x hx => ?_⟩
dsimp only
rw [hx]
| 2,121 |
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Order.Filter.IndicatorFunction
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.LpSeminorm.Trim
#align_import measure_theory.function.conditional_expectation.ae_measurable from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
set_option linter.uppercaseLean3 false
open TopologicalSpace Filter
open scoped ENNReal MeasureTheory
namespace MeasureTheory
def AEStronglyMeasurable' {α β} [TopologicalSpace β] (m : MeasurableSpace α)
{_ : MeasurableSpace α} (f : α → β) (μ : Measure α) : Prop :=
∃ g : α → β, StronglyMeasurable[m] g ∧ f =ᵐ[μ] g
#align measure_theory.ae_strongly_measurable' MeasureTheory.AEStronglyMeasurable'
| Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 136 | 139 | theorem aeStronglyMeasurable'_of_aeStronglyMeasurable'_trim {α β} {m m0 m0' : MeasurableSpace α}
[TopologicalSpace β] (hm0 : m0 ≤ m0') {μ : Measure α} {f : α → β}
(hf : AEStronglyMeasurable' m f (μ.trim hm0)) : AEStronglyMeasurable' m f μ := by |
obtain ⟨g, hg_meas, hfg⟩ := hf; exact ⟨g, hg_meas, ae_eq_of_ae_eq_trim hfg⟩
| 2,121 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputable def kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
AddSubmonoid (α → Measure β) where
carrier := Measurable
zero_mem' := measurable_zero
add_mem' hf hg := Measurable.add hf hg
#align probability_theory.kernel ProbabilityTheory.kernel
-- Porting note: using `FunLike` instead of `CoeFun` to use `DFunLike.coe`
instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
FunLike (kernel α β) α (Measure β) where
coe := Subtype.val
coe_injective' := Subtype.val_injective
instance kernel.instCovariantAddLE {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
CovariantClass (kernel α β) (kernel α β) (· + ·) (· ≤ ·) :=
⟨fun _ _ _ hμ a ↦ add_le_add_left (hμ a) _⟩
noncomputable
instance kernel.instOrderBot {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
OrderBot (kernel α β) where
bot := 0
bot_le κ a := by simp only [ZeroMemClass.coe_zero, Pi.zero_apply, Measure.zero_le]
variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
namespace kernel
@[simp]
theorem coeFn_zero : ⇑(0 : kernel α β) = 0 :=
rfl
#align probability_theory.kernel.coe_fn_zero ProbabilityTheory.kernel.coeFn_zero
@[simp]
theorem coeFn_add (κ η : kernel α β) : ⇑(κ + η) = κ + η :=
rfl
#align probability_theory.kernel.coe_fn_add ProbabilityTheory.kernel.coeFn_add
def coeAddHom (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
kernel α β →+ α → Measure β :=
AddSubmonoid.subtype _
#align probability_theory.kernel.coe_add_hom ProbabilityTheory.kernel.coeAddHom
@[simp]
theorem zero_apply (a : α) : (0 : kernel α β) a = 0 :=
rfl
#align probability_theory.kernel.zero_apply ProbabilityTheory.kernel.zero_apply
@[simp]
theorem coe_finset_sum (I : Finset ι) (κ : ι → kernel α β) : ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i) :=
map_sum (coeAddHom α β) _ _
#align probability_theory.kernel.coe_finset_sum ProbabilityTheory.kernel.coe_finset_sum
| Mathlib/Probability/Kernel/Basic.lean | 113 | 114 | theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :
(∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by | rw [coe_finset_sum, Finset.sum_apply]
| 2,122 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputable def kernel (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
AddSubmonoid (α → Measure β) where
carrier := Measurable
zero_mem' := measurable_zero
add_mem' hf hg := Measurable.add hf hg
#align probability_theory.kernel ProbabilityTheory.kernel
-- Porting note: using `FunLike` instead of `CoeFun` to use `DFunLike.coe`
instance {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
FunLike (kernel α β) α (Measure β) where
coe := Subtype.val
coe_injective' := Subtype.val_injective
instance kernel.instCovariantAddLE {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
CovariantClass (kernel α β) (kernel α β) (· + ·) (· ≤ ·) :=
⟨fun _ _ _ hμ a ↦ add_le_add_left (hμ a) _⟩
noncomputable
instance kernel.instOrderBot {α β : Type*} [MeasurableSpace α] [MeasurableSpace β] :
OrderBot (kernel α β) where
bot := 0
bot_le κ a := by simp only [ZeroMemClass.coe_zero, Pi.zero_apply, Measure.zero_le]
variable {α β ι : Type*} {mα : MeasurableSpace α} {mβ : MeasurableSpace β}
namespace kernel
@[simp]
theorem coeFn_zero : ⇑(0 : kernel α β) = 0 :=
rfl
#align probability_theory.kernel.coe_fn_zero ProbabilityTheory.kernel.coeFn_zero
@[simp]
theorem coeFn_add (κ η : kernel α β) : ⇑(κ + η) = κ + η :=
rfl
#align probability_theory.kernel.coe_fn_add ProbabilityTheory.kernel.coeFn_add
def coeAddHom (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] :
kernel α β →+ α → Measure β :=
AddSubmonoid.subtype _
#align probability_theory.kernel.coe_add_hom ProbabilityTheory.kernel.coeAddHom
@[simp]
theorem zero_apply (a : α) : (0 : kernel α β) a = 0 :=
rfl
#align probability_theory.kernel.zero_apply ProbabilityTheory.kernel.zero_apply
@[simp]
theorem coe_finset_sum (I : Finset ι) (κ : ι → kernel α β) : ⇑(∑ i ∈ I, κ i) = ∑ i ∈ I, ⇑(κ i) :=
map_sum (coeAddHom α β) _ _
#align probability_theory.kernel.coe_finset_sum ProbabilityTheory.kernel.coe_finset_sum
theorem finset_sum_apply (I : Finset ι) (κ : ι → kernel α β) (a : α) :
(∑ i ∈ I, κ i) a = ∑ i ∈ I, κ i a := by rw [coe_finset_sum, Finset.sum_apply]
#align probability_theory.kernel.finset_sum_apply ProbabilityTheory.kernel.finset_sum_apply
| Mathlib/Probability/Kernel/Basic.lean | 117 | 118 | theorem finset_sum_apply' (I : Finset ι) (κ : ι → kernel α β) (a : α) (s : Set β) :
(∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by | rw [finset_sum_apply, Measure.finset_sum_apply]
| 2,122 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
| Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 28 | 41 | theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :
∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc
_ = ∫ a in p.support, f a ∂(p.toMeasure) := by | rw [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by
apply integral_countable f p.support_countable
rwa [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p a).toReal • f a := by
congr with x; congr 2
apply PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
_ = ∑' a, (p a).toReal • f a :=
tsum_subtype_eq_of_support_subset <| by calc
(fun a ↦ (p a).toReal • f a).support ⊆ (fun a ↦ (p a).toReal).support :=
Function.support_smul_subset_left _ _
_ ⊆ support p := fun x h1 h2 => h1 (by simp [h2])
| 2,123 |
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {α : Type*} [MeasurableSpace α] [MeasurableSingletonClass α]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
theorem integral_eq_tsum (p : PMF α) (f : α → E) (hf : Integrable f p.toMeasure) :
∫ a, f a ∂(p.toMeasure) = ∑' a, (p a).toReal • f a := calc
_ = ∫ a in p.support, f a ∂(p.toMeasure) := by rw [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p.toMeasure {a.val}).toReal • f a := by
apply integral_countable f p.support_countable
rwa [restrict_toMeasure_support p]
_ = ∑' (a : support p), (p a).toReal • f a := by
congr with x; congr 2
apply PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
_ = ∑' a, (p a).toReal • f a :=
tsum_subtype_eq_of_support_subset <| by calc
(fun a ↦ (p a).toReal • f a).support ⊆ (fun a ↦ (p a).toReal).support :=
Function.support_smul_subset_left _ _
_ ⊆ support p := fun x h1 h2 => h1 (by simp [h2])
| Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 43 | 47 | theorem integral_eq_sum [Fintype α] (p : PMF α) (f : α → E) :
∫ a, f a ∂(p.toMeasure) = ∑ a, (p a).toReal • f a := by |
rw [integral_fintype _ (.of_finite _ f)]
congr with x; congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
| 2,123 |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureTheory.Integral.Bochner
#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"e0736bb5b48bdadbca19dbd857e12bee38ccfbb8"
open scoped ENNReal NNReal Topology BoundedContinuousFunction
open MeasureTheory TopologicalSpace ContinuousMap Set Bornology
variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [T4Space α] [BorelSpace α]
variable {E : Type*} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
namespace MeasureTheory
variable [NormedSpace ℝ E]
| Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 78 | 134 | theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
(s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
(hε : ε ≠ 0) :
∃ f : α → E,
Continuous f ∧
snorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
(∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ Memℒp f p μ := by |
obtain ⟨η, η_pos, hη⟩ :
∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ ε :=
exists_snorm_indicator_le hp c hε
have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η :=
s_closed.measurableSet.exists_isOpen_diff_lt hs ηpos.ne'
let v := u ∩ V
have hsv : s ⊆ v := subset_inter hsu sV
have hμv : μ v < ∞ := (measure_mono inter_subset_right).trans_lt h'V
obtain ⟨g, hgv, hgs, hg_range⟩ :=
exists_continuous_zero_one_of_isClosed (u_open.inter V_open).isClosed_compl s_closed
(disjoint_compl_left_iff.2 hsv)
-- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
-- that this is pointwise bounded by the indicator of the set `v \ s`, which has small measure.
have g_norm : ∀ x, ‖g x‖ = g x := fun x => by rw [Real.norm_eq_abs, abs_of_nonneg (hg_range x).1]
have gc_bd0 : ∀ x, ‖g x • c‖ ≤ ‖c‖ := by
intro x
simp only [norm_smul, g_norm x]
apply mul_le_of_le_one_left (norm_nonneg _)
exact (hg_range x).2
have gc_bd :
∀ x, ‖g x • c - s.indicator (fun _x => c) x‖ ≤ ‖(v \ s).indicator (fun _x => c) x‖ := by
intro x
by_cases hv : x ∈ v
· rw [← Set.diff_union_of_subset hsv] at hv
cases' hv with hsv hs
· simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
Set.indicator_of_mem] using gc_bd0 x
· simp [hgs hs, hs]
· simp [hgv hv, show x ∉ s from fun h => hv (hsv h)]
have gc_support : (Function.support fun x : α => g x • c) ⊆ v := by
refine Function.support_subset_iff'.2 fun x hx => ?_
simp only [hgv hx, Pi.zero_apply, zero_smul]
have gc_mem : Memℒp (fun x => g x • c) p μ := by
refine Memℒp.smul_of_top_left (memℒp_top_const _) ?_
refine ⟨g.continuous.aestronglyMeasurable, ?_⟩
have : snorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ := by
refine (snorm_indicator_const_le _ _).trans_lt ?_
simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne,
ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and_iff, or_false_iff, not_and, not_lt,
ENNReal.toReal_nonneg, imp_true_iff]
refine (snorm_mono fun x => ?_).trans_lt this
by_cases hx : x ∈ v
· simp only [hx, abs_of_nonneg (hg_range x).1, (hg_range x).2, Real.norm_eq_abs,
indicator_of_mem, CstarRing.norm_one]
· simp only [hgv hx, Pi.zero_apply, Real.norm_eq_abs, abs_zero, abs_nonneg]
refine
⟨fun x => g x • c, g.continuous.smul continuous_const, (snorm_mono gc_bd).trans ?_, gc_bd0,
gc_support.trans inter_subset_left, gc_mem⟩
exact hη _ ((measure_mono (diff_subset_diff inter_subset_right Subset.rfl)).trans hV.le)
| 2,124 |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
import Mathlib.Topology.UrysohnsLemma
import Mathlib.MeasureTheory.Integral.Bochner
#align_import measure_theory.function.continuous_map_dense from "leanprover-community/mathlib"@"e0736bb5b48bdadbca19dbd857e12bee38ccfbb8"
open scoped ENNReal NNReal Topology BoundedContinuousFunction
open MeasureTheory TopologicalSpace ContinuousMap Set Bornology
variable {α : Type*} [MeasurableSpace α] [TopologicalSpace α] [T4Space α] [BorelSpace α]
variable {E : Type*} [NormedAddCommGroup E] {μ : Measure α} {p : ℝ≥0∞}
namespace MeasureTheory
variable [NormedSpace ℝ E]
theorem exists_continuous_snorm_sub_le_of_closed [μ.OuterRegular] (hp : p ≠ ∞) {s u : Set α}
(s_closed : IsClosed s) (u_open : IsOpen u) (hsu : s ⊆ u) (hs : μ s ≠ ∞) (c : E) {ε : ℝ≥0∞}
(hε : ε ≠ 0) :
∃ f : α → E,
Continuous f ∧
snorm (fun x => f x - s.indicator (fun _y => c) x) p μ ≤ ε ∧
(∀ x, ‖f x‖ ≤ ‖c‖) ∧ Function.support f ⊆ u ∧ Memℒp f p μ := by
obtain ⟨η, η_pos, hη⟩ :
∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ ε :=
exists_snorm_indicator_le hp c hε
have ηpos : (0 : ℝ≥0∞) < η := ENNReal.coe_lt_coe.2 η_pos
obtain ⟨V, sV, V_open, h'V, hV⟩ : ∃ (V : Set α), V ⊇ s ∧ IsOpen V ∧ μ V < ∞ ∧ μ (V \ s) < η :=
s_closed.measurableSet.exists_isOpen_diff_lt hs ηpos.ne'
let v := u ∩ V
have hsv : s ⊆ v := subset_inter hsu sV
have hμv : μ v < ∞ := (measure_mono inter_subset_right).trans_lt h'V
obtain ⟨g, hgv, hgs, hg_range⟩ :=
exists_continuous_zero_one_of_isClosed (u_open.inter V_open).isClosed_compl s_closed
(disjoint_compl_left_iff.2 hsv)
-- Multiply this by `c` to get a continuous approximation to the function `f`; the key point is
-- that this is pointwise bounded by the indicator of the set `v \ s`, which has small measure.
have g_norm : ∀ x, ‖g x‖ = g x := fun x => by rw [Real.norm_eq_abs, abs_of_nonneg (hg_range x).1]
have gc_bd0 : ∀ x, ‖g x • c‖ ≤ ‖c‖ := by
intro x
simp only [norm_smul, g_norm x]
apply mul_le_of_le_one_left (norm_nonneg _)
exact (hg_range x).2
have gc_bd :
∀ x, ‖g x • c - s.indicator (fun _x => c) x‖ ≤ ‖(v \ s).indicator (fun _x => c) x‖ := by
intro x
by_cases hv : x ∈ v
· rw [← Set.diff_union_of_subset hsv] at hv
cases' hv with hsv hs
· simpa only [hsv.2, Set.indicator_of_not_mem, not_false_iff, sub_zero, hsv,
Set.indicator_of_mem] using gc_bd0 x
· simp [hgs hs, hs]
· simp [hgv hv, show x ∉ s from fun h => hv (hsv h)]
have gc_support : (Function.support fun x : α => g x • c) ⊆ v := by
refine Function.support_subset_iff'.2 fun x hx => ?_
simp only [hgv hx, Pi.zero_apply, zero_smul]
have gc_mem : Memℒp (fun x => g x • c) p μ := by
refine Memℒp.smul_of_top_left (memℒp_top_const _) ?_
refine ⟨g.continuous.aestronglyMeasurable, ?_⟩
have : snorm (v.indicator fun _x => (1 : ℝ)) p μ < ⊤ := by
refine (snorm_indicator_const_le _ _).trans_lt ?_
simp only [lt_top_iff_ne_top, hμv.ne, nnnorm_one, ENNReal.coe_one, one_div, one_mul, Ne,
ENNReal.rpow_eq_top_iff, inv_lt_zero, false_and_iff, or_false_iff, not_and, not_lt,
ENNReal.toReal_nonneg, imp_true_iff]
refine (snorm_mono fun x => ?_).trans_lt this
by_cases hx : x ∈ v
· simp only [hx, abs_of_nonneg (hg_range x).1, (hg_range x).2, Real.norm_eq_abs,
indicator_of_mem, CstarRing.norm_one]
· simp only [hgv hx, Pi.zero_apply, Real.norm_eq_abs, abs_zero, abs_nonneg]
refine
⟨fun x => g x • c, g.continuous.smul continuous_const, (snorm_mono gc_bd).trans ?_, gc_bd0,
gc_support.trans inter_subset_left, gc_mem⟩
exact hη _ ((measure_mono (diff_subset_diff inter_subset_right Subset.rfl)).trans hV.le)
#align measure_theory.exists_continuous_snorm_sub_le_of_closed MeasureTheory.exists_continuous_snorm_sub_le_of_closed
| Mathlib/MeasureTheory/Function/ContinuousMapDense.lean | 139 | 188 | theorem Memℒp.exists_hasCompactSupport_snorm_sub_le [WeaklyLocallyCompactSpace α] [μ.Regular]
(hp : p ≠ ∞) {f : α → E} (hf : Memℒp f p μ) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ g : α → E, HasCompactSupport g ∧ snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ := by |
suffices H :
∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ HasCompactSupport g by
rcases H with ⟨g, hg, g_cont, g_mem, g_support⟩
exact ⟨g, g_support, hg, g_cont, g_mem⟩
-- It suffices to check that the set of functions we consider approximates characteristic
-- functions, is stable under addition and consists of ae strongly measurable functions.
-- First check the latter easy facts.
apply hf.induction_dense hp _ _ _ _ hε
rotate_left
-- stability under addition
· rintro f g ⟨f_cont, f_mem, hf⟩ ⟨g_cont, g_mem, hg⟩
exact ⟨f_cont.add g_cont, f_mem.add g_mem, hf.add hg⟩
-- ae strong measurability
· rintro f ⟨_f_cont, f_mem, _hf⟩
exact f_mem.aestronglyMeasurable
-- We are left with approximating characteristic functions.
-- This follows from `exists_continuous_snorm_sub_le_of_closed`.
intro c t ht htμ ε hε
rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩
obtain ⟨η, ηpos, hη⟩ :
∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ :=
exists_snorm_indicator_le hp c δpos.ne'
have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos
obtain ⟨s, st, s_compact, μs⟩ : ∃ s, s ⊆ t ∧ IsCompact s ∧ μ (t \ s) < η :=
ht.exists_isCompact_diff_lt htμ.ne hη_pos'.ne'
have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ
have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ := by
rw [← snorm_neg, neg_sub, ← indicator_diff st]
exact hη _ μs.le
obtain ⟨k, k_compact, sk⟩ : ∃ k : Set α, IsCompact k ∧ s ⊆ interior k :=
exists_compact_superset s_compact
rcases exists_continuous_snorm_sub_le_of_closed hp s_compact.isClosed isOpen_interior sk hsμ.ne c
δpos.ne' with
⟨f, f_cont, I2, _f_bound, f_support, f_mem⟩
have I3 : snorm (f - t.indicator fun _y => c) p μ ≤ ε := by
convert
(hδ _ _
(f_mem.aestronglyMeasurable.sub
(aestronglyMeasurable_const.indicator s_compact.measurableSet))
((aestronglyMeasurable_const.indicator s_compact.measurableSet).sub
(aestronglyMeasurable_const.indicator ht))
I2 I1).le using 2
simp only [sub_add_sub_cancel]
refine ⟨f, I3, f_cont, f_mem, HasCompactSupport.intro k_compact fun x hx => ?_⟩
rw [← Function.nmem_support]
contrapose! hx
exact interior_subset (f_support hx)
| 2,124 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞}
section SMul
variable [SMul M α] [SMulInvariantMeasure M α μ] [MeasurableSMul M α]
@[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) where
smul c f := Lp.compMeasurePreserving (mk.symm c • ·) (measurePreserving_smul _ _) f
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 := rfl
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) :=
Lp.coeFn_compMeasurePreserving _ _
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
mk c • hf.toLp f =
(hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) :=
rfl
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
c • Lp.const p μ a = Lp.const p μ a :=
rfl
instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
-- We don't have a typeclass for additive versions of the next few lemmas
-- Should we add `AddDistribAddAction` with `to_additive` both from `MulDistribMulAction`
-- and `DistribMulAction`?
@[to_additive]
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 70 | 71 | theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by |
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
| 2,125 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞}
section SMul
variable [SMul M α] [SMulInvariantMeasure M α μ] [MeasurableSMul M α]
@[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) where
smul c f := Lp.compMeasurePreserving (mk.symm c • ·) (measurePreserving_smul _ _) f
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 := rfl
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) :=
Lp.coeFn_compMeasurePreserving _ _
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
mk c • hf.toLp f =
(hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) :=
rfl
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
c • Lp.const p μ a = Lp.const p μ a :=
rfl
instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
-- We don't have a typeclass for additive versions of the next few lemmas
-- Should we add `AddDistribAddAction` with `to_additive` both from `MulDistribMulAction`
-- and `DistribMulAction`?
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
attribute [simp] DomAddAct.vadd_Lp_add
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 := rfl
@[to_additive]
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 78 | 79 | theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) := by |
rcases f with ⟨⟨_⟩, _⟩; rfl
| 2,125 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞}
section SMul
variable [SMul M α] [SMulInvariantMeasure M α μ] [MeasurableSMul M α]
@[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) where
smul c f := Lp.compMeasurePreserving (mk.symm c • ·) (measurePreserving_smul _ _) f
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 := rfl
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) :=
Lp.coeFn_compMeasurePreserving _ _
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
mk c • hf.toLp f =
(hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) :=
rfl
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
c • Lp.const p μ a = Lp.const p μ a :=
rfl
instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
-- We don't have a typeclass for additive versions of the next few lemmas
-- Should we add `AddDistribAddAction` with `to_additive` both from `MulDistribMulAction`
-- and `DistribMulAction`?
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
attribute [simp] DomAddAct.vadd_Lp_add
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 := rfl
@[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) := by
rcases f with ⟨⟨_⟩, _⟩; rfl
@[to_additive]
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 82 | 83 | theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g := by |
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
| 2,125 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞}
section SMul
variable [SMul M α] [SMulInvariantMeasure M α μ] [MeasurableSMul M α]
@[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) where
smul c f := Lp.compMeasurePreserving (mk.symm c • ·) (measurePreserving_smul _ _) f
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 := rfl
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) :=
Lp.coeFn_compMeasurePreserving _ _
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
mk c • hf.toLp f =
(hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) :=
rfl
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
c • Lp.const p μ a = Lp.const p μ a :=
rfl
instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
-- We don't have a typeclass for additive versions of the next few lemmas
-- Should we add `AddDistribAddAction` with `to_additive` both from `MulDistribMulAction`
-- and `DistribMulAction`?
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
attribute [simp] DomAddAct.vadd_Lp_add
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 := rfl
@[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) := by
rcases f with ⟨⟨_⟩, _⟩; rfl
@[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
instance : DistribSMul Mᵈᵐᵃ (Lp E p μ) where
smul_zero _ := rfl
smul_add := by rintro _ ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
-- The next few lemmas follow from the `IsometricSMul` instance if `1 ≤ p`
@[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ :=
Lp.norm_compMeasurePreserving _ _
@[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ :=
NNReal.eq <| Lp.norm_compMeasurePreserving _ _
@[to_additive (attr := simp)]
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 99 | 100 | theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g := by |
simp only [dist, ← smul_Lp_sub, norm_smul_Lp]
| 2,125 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : MeasureTheory.Measure α} {p : ℝ≥0∞}
section SMul
variable [SMul M α] [SMulInvariantMeasure M α μ] [MeasurableSMul M α]
@[to_additive]
instance : SMul Mᵈᵐᵃ (Lp E p μ) where
smul c f := Lp.compMeasurePreserving (mk.symm c • ·) (measurePreserving_smul _ _) f
@[to_additive (attr := simp)]
theorem smul_Lp_val (c : Mᵈᵐᵃ) (f : Lp E p μ) : (c • f).1 = c • f.1 := rfl
@[to_additive]
theorem smul_Lp_ae_eq (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • f =ᵐ[μ] (f <| mk.symm c • ·) :=
Lp.coeFn_compMeasurePreserving _ _
@[to_additive]
theorem mk_smul_toLp (c : M) {f : α → E} (hf : Memℒp f p μ) :
mk c • hf.toLp f =
(hf.comp_measurePreserving <| measurePreserving_smul c μ).toLp (f <| c • ·) :=
rfl
@[to_additive (attr := simp)]
theorem smul_Lp_const [IsFiniteMeasure μ] (c : Mᵈᵐᵃ) (a : E) :
c • Lp.const p μ a = Lp.const p μ a :=
rfl
instance [SMul N α] [SMulCommClass M N α] [SMulInvariantMeasure N α μ] [MeasurableSMul N α] :
SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass Mᵈᵐᵃ 𝕜 (Lp E p μ) :=
Subtype.val_injective.smulCommClass (fun _ _ ↦ rfl) fun _ _ ↦ rfl
instance [NormedRing 𝕜] [Module 𝕜 E] [BoundedSMul 𝕜 E] : SMulCommClass 𝕜 Mᵈᵐᵃ (Lp E p μ) :=
.symm _ _ _
-- We don't have a typeclass for additive versions of the next few lemmas
-- Should we add `AddDistribAddAction` with `to_additive` both from `MulDistribMulAction`
-- and `DistribMulAction`?
@[to_additive]
theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
attribute [simp] DomAddAct.vadd_Lp_add
@[to_additive (attr := simp 1001)]
theorem smul_Lp_zero (c : Mᵈᵐᵃ) : c • (0 : Lp E p μ) = 0 := rfl
@[to_additive]
theorem smul_Lp_neg (c : Mᵈᵐᵃ) (f : Lp E p μ) : c • (-f) = -(c • f) := by
rcases f with ⟨⟨_⟩, _⟩; rfl
@[to_additive]
theorem smul_Lp_sub (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f - g) = c • f - c • g := by
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
instance : DistribSMul Mᵈᵐᵃ (Lp E p μ) where
smul_zero _ := rfl
smul_add := by rintro _ ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
-- The next few lemmas follow from the `IsometricSMul` instance if `1 ≤ p`
@[to_additive (attr := simp)]
theorem norm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖ = ‖f‖ :=
Lp.norm_compMeasurePreserving _ _
@[to_additive (attr := simp)]
theorem nnnorm_smul_Lp (c : Mᵈᵐᵃ) (f : Lp E p μ) : ‖c • f‖₊ = ‖f‖₊ :=
NNReal.eq <| Lp.norm_compMeasurePreserving _ _
@[to_additive (attr := simp)]
theorem dist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : dist (c • f) (c • g) = dist f g := by
simp only [dist, ← smul_Lp_sub, norm_smul_Lp]
@[to_additive (attr := simp)]
| Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 103 | 104 | theorem edist_smul_Lp (c : Mᵈᵐᵃ) (f g : Lp E p μ) : edist (c • f) (c • g) = edist f g := by |
simp only [Lp.edist_dist, dist_smul_Lp]
| 2,125 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
| Mathlib/Analysis/Convex/Normed.lean | 39 | 44 | theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by |
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
| Mathlib/Analysis/Convex/Normed.lean | 53 | 55 | theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by |
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
| Mathlib/Analysis/Convex/Normed.lean | 62 | 63 | theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by |
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
| Mathlib/Analysis/Convex/Normed.lean | 66 | 67 | theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by |
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
| Mathlib/Analysis/Convex/Normed.lean | 70 | 72 | theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by |
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
| Mathlib/Analysis/Convex/Normed.lean | 75 | 80 | theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by |
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
#align convex.cthickening Convex.cthickening
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y :=
(convexOn_dist y (convex_convexHull ℝ _)).exists_ge_of_mem_convexHull hx
#align convex_hull_exists_dist_ge convexHull_exists_dist_ge
| Mathlib/Analysis/Convex/Normed.lean | 92 | 97 | theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by |
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
#align convex.cthickening Convex.cthickening
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y :=
(convexOn_dist y (convex_convexHull ℝ _)).exists_ge_of_mem_convexHull hx
#align convex_hull_exists_dist_ge convexHull_exists_dist_ge
theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
#align convex_hull_exists_dist_ge2 convexHull_exists_dist_ge2
@[simp]
| Mathlib/Analysis/Convex/Normed.lean | 102 | 108 | theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s := by |
refine (EMetric.diam_le fun x hx y hy => ?_).antisymm (EMetric.diam_mono <| subset_convexHull ℝ s)
rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
rw [edist_dist]
apply le_trans (ENNReal.ofReal_le_ofReal H)
rw [← edist_dist]
exact EMetric.edist_le_diam_of_mem hx' hy'
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
#align convex.cthickening Convex.cthickening
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y :=
(convexOn_dist y (convex_convexHull ℝ _)).exists_ge_of_mem_convexHull hx
#align convex_hull_exists_dist_ge convexHull_exists_dist_ge
theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
#align convex_hull_exists_dist_ge2 convexHull_exists_dist_ge2
@[simp]
theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s := by
refine (EMetric.diam_le fun x hx y hy => ?_).antisymm (EMetric.diam_mono <| subset_convexHull ℝ s)
rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
rw [edist_dist]
apply le_trans (ENNReal.ofReal_le_ofReal H)
rw [← edist_dist]
exact EMetric.edist_le_diam_of_mem hx' hy'
#align convex_hull_ediam convexHull_ediam
@[simp]
| Mathlib/Analysis/Convex/Normed.lean | 113 | 114 | theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s := by |
simp only [Metric.diam, convexHull_ediam]
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
#align convex.cthickening Convex.cthickening
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y :=
(convexOn_dist y (convex_convexHull ℝ _)).exists_ge_of_mem_convexHull hx
#align convex_hull_exists_dist_ge convexHull_exists_dist_ge
theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
#align convex_hull_exists_dist_ge2 convexHull_exists_dist_ge2
@[simp]
theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s := by
refine (EMetric.diam_le fun x hx y hy => ?_).antisymm (EMetric.diam_mono <| subset_convexHull ℝ s)
rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
rw [edist_dist]
apply le_trans (ENNReal.ofReal_le_ofReal H)
rw [← edist_dist]
exact EMetric.edist_le_diam_of_mem hx' hy'
#align convex_hull_ediam convexHull_ediam
@[simp]
theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s := by
simp only [Metric.diam, convexHull_ediam]
#align convex_hull_diam convexHull_diam
@[simp]
| Mathlib/Analysis/Convex/Normed.lean | 119 | 121 | theorem isBounded_convexHull {s : Set E} :
Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s := by |
simp only [Metric.isBounded_iff_ediam_ne_top, convexHull_ediam]
| 2,126 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {ι : Type*} {E P : Type*}
open Metric Set
open scoped Convex
variable [SeminormedAddCommGroup E] [NormedSpace ℝ E] [PseudoMetricSpace P] [NormedAddTorsor E P]
variable {s t : Set E}
theorem convexOn_norm (hs : Convex ℝ s) : ConvexOn ℝ s norm :=
⟨hs, fun x _ y _ a b ha hb _ =>
calc
‖a • x + b • y‖ ≤ ‖a • x‖ + ‖b • y‖ := norm_add_le _ _
_ = a * ‖x‖ + b * ‖y‖ := by
rw [norm_smul, norm_smul, Real.norm_of_nonneg ha, Real.norm_of_nonneg hb]⟩
#align convex_on_norm convexOn_norm
theorem convexOn_univ_norm : ConvexOn ℝ univ (norm : E → ℝ) :=
convexOn_norm convex_univ
#align convex_on_univ_norm convexOn_univ_norm
theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
#align convex_on_dist convexOn_dist
theorem convexOn_univ_dist (z : E) : ConvexOn ℝ univ fun z' => dist z' z :=
convexOn_dist z convex_univ
#align convex_on_univ_dist convexOn_univ_dist
theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
#align convex_ball convex_ball
theorem convex_closedBall (a : E) (r : ℝ) : Convex ℝ (Metric.closedBall a r) := by
simpa only [Metric.closedBall, sep_univ] using (convexOn_univ_dist a).convex_le r
#align convex_closed_ball convex_closedBall
theorem Convex.thickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (thickening δ s) := by
rw [← add_ball_zero]
exact hs.add (convex_ball 0 _)
#align convex.thickening Convex.thickening
theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
#align convex.cthickening Convex.cthickening
theorem convexHull_exists_dist_ge {s : Set E} {x : E} (hx : x ∈ convexHull ℝ s) (y : E) :
∃ x' ∈ s, dist x y ≤ dist x' y :=
(convexOn_dist y (convex_convexHull ℝ _)).exists_ge_of_mem_convexHull hx
#align convex_hull_exists_dist_ge convexHull_exists_dist_ge
theorem convexHull_exists_dist_ge2 {s t : Set E} {x y : E} (hx : x ∈ convexHull ℝ s)
(hy : y ∈ convexHull ℝ t) : ∃ x' ∈ s, ∃ y' ∈ t, dist x y ≤ dist x' y' := by
rcases convexHull_exists_dist_ge hx y with ⟨x', hx', Hx'⟩
rcases convexHull_exists_dist_ge hy x' with ⟨y', hy', Hy'⟩
use x', hx', y', hy'
exact le_trans Hx' (dist_comm y x' ▸ dist_comm y' x' ▸ Hy')
#align convex_hull_exists_dist_ge2 convexHull_exists_dist_ge2
@[simp]
theorem convexHull_ediam (s : Set E) : EMetric.diam (convexHull ℝ s) = EMetric.diam s := by
refine (EMetric.diam_le fun x hx y hy => ?_).antisymm (EMetric.diam_mono <| subset_convexHull ℝ s)
rcases convexHull_exists_dist_ge2 hx hy with ⟨x', hx', y', hy', H⟩
rw [edist_dist]
apply le_trans (ENNReal.ofReal_le_ofReal H)
rw [← edist_dist]
exact EMetric.edist_le_diam_of_mem hx' hy'
#align convex_hull_ediam convexHull_ediam
@[simp]
theorem convexHull_diam (s : Set E) : Metric.diam (convexHull ℝ s) = Metric.diam s := by
simp only [Metric.diam, convexHull_ediam]
#align convex_hull_diam convexHull_diam
@[simp]
theorem isBounded_convexHull {s : Set E} :
Bornology.IsBounded (convexHull ℝ s) ↔ Bornology.IsBounded s := by
simp only [Metric.isBounded_iff_ediam_ne_top, convexHull_ediam]
#align bounded_convex_hull isBounded_convexHull
instance (priority := 100) NormedSpace.instPathConnectedSpace : PathConnectedSpace E :=
TopologicalAddGroup.pathConnectedSpace
#align normed_space.path_connected NormedSpace.instPathConnectedSpace
instance (priority := 100) NormedSpace.instLocPathConnectedSpace : LocPathConnectedSpace E :=
locPathConnected_of_bases (fun x => Metric.nhds_basis_ball) fun x r r_pos =>
(convex_ball x r).isPathConnected <| by simp [r_pos]
#align normed_space.loc_path_connected NormedSpace.instLocPathConnectedSpace
| Mathlib/Analysis/Convex/Normed.lean | 133 | 136 | theorem Wbtw.dist_add_dist {x y z : P} (h : Wbtw ℝ x y z) :
dist x y + dist y z = dist x z := by |
obtain ⟨a, ⟨ha₀, ha₁⟩, rfl⟩ := h
simp [abs_of_nonneg, ha₀, ha₁, sub_mul]
| 2,126 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 76 | 81 | theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by |
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
| 2,127 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
#align strict_convex_closed_ball strictConvex_closedBall
variable [NormedSpace ℝ E]
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
(h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E :=
⟨fun r hr => by simpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r⟩
#align strict_convex_space.of_strict_convex_closed_unit_ball StrictConvexSpace.of_strictConvex_closed_unit_ball
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 95 | 106 | theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by |
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,
sub_eq_iff_eq_add.2 hab.symm]
| 2,127 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
#align strict_convex_closed_ball strictConvex_closedBall
variable [NormedSpace ℝ E]
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
(h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E :=
⟨fun r hr => by simpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r⟩
#align strict_convex_space.of_strict_convex_closed_unit_ball StrictConvexSpace.of_strictConvex_closed_unit_ball
theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,
sub_eq_iff_eq_add.2 hab.symm]
#align strict_convex_space.of_norm_combo_lt_one StrictConvexSpace.of_norm_combo_lt_one
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 109 | 120 | theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with ⟨a, b, ha, hb, hab, hne'⟩
exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩
| 2,127 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
#align strict_convex_closed_ball strictConvex_closedBall
variable [NormedSpace ℝ E]
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
(h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E :=
⟨fun r hr => by simpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r⟩
#align strict_convex_space.of_strict_convex_closed_unit_ball StrictConvexSpace.of_strictConvex_closed_unit_ball
theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,
sub_eq_iff_eq_add.2 hab.symm]
#align strict_convex_space.of_norm_combo_lt_one StrictConvexSpace.of_norm_combo_lt_one
theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with ⟨a, b, ha, hb, hab, hne'⟩
exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩
#align strict_convex_space.of_norm_combo_ne_one StrictConvexSpace.of_norm_combo_ne_one
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 123 | 130 | theorem StrictConvexSpace.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E := by |
refine
StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>
⟨1 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, ?_⟩
rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' ℝ)]
exact h hx hy hne
| 2,127 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
#align strict_convex_closed_ball strictConvex_closedBall
variable [NormedSpace ℝ E]
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
(h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E :=
⟨fun r hr => by simpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r⟩
#align strict_convex_space.of_strict_convex_closed_unit_ball StrictConvexSpace.of_strictConvex_closed_unit_ball
theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,
sub_eq_iff_eq_add.2 hab.symm]
#align strict_convex_space.of_norm_combo_lt_one StrictConvexSpace.of_norm_combo_lt_one
theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with ⟨a, b, ha, hb, hab, hne'⟩
exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩
#align strict_convex_space.of_norm_combo_ne_one StrictConvexSpace.of_norm_combo_ne_one
theorem StrictConvexSpace.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>
⟨1 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, ?_⟩
rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' ℝ)]
exact h hx hy hne
#align strict_convex_space.of_norm_add_ne_two StrictConvexSpace.of_norm_add_ne_two
theorem StrictConvexSpace.of_pairwise_sphere_norm_ne_two
(h : (sphere (0 : E) 1).Pairwise fun x y => ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E :=
StrictConvexSpace.of_norm_add_ne_two fun _ _ hx hy =>
h (mem_sphere_zero_iff_norm.2 hx) (mem_sphere_zero_iff_norm.2 hy)
#align strict_convex_space.of_pairwise_sphere_norm_ne_two StrictConvexSpace.of_pairwise_sphere_norm_ne_two
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 141 | 145 | theorem StrictConvexSpace.of_norm_add
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y) : StrictConvexSpace ℝ E := by |
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h₂ => ?_
rw [mem_sphere_zero_iff_norm] at hx hy
exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)
| 2,127 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
open Convex Pointwise Set Metric
class StrictConvexSpace (𝕜 E : Type*) [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E] : Prop where
strictConvex_closedBall : ∀ r : ℝ, 0 < r → StrictConvex 𝕜 (closedBall (0 : E) r)
#align strict_convex_space StrictConvexSpace
variable (𝕜 : Type*) {E : Type*} [NormedLinearOrderedField 𝕜] [NormedAddCommGroup E]
[NormedSpace 𝕜 E]
theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
#align strict_convex_closed_ball strictConvex_closedBall
variable [NormedSpace ℝ E]
theorem StrictConvexSpace.of_strictConvex_closed_unit_ball [LinearMap.CompatibleSMul E E 𝕜 ℝ]
(h : StrictConvex 𝕜 (closedBall (0 : E) 1)) : StrictConvexSpace 𝕜 E :=
⟨fun r hr => by simpa only [smul_closedUnitBall_of_nonneg hr.le] using h.smul r⟩
#align strict_convex_space.of_strict_convex_closed_unit_ball StrictConvexSpace.of_strictConvex_closed_unit_ball
theorem StrictConvexSpace.of_norm_combo_lt_one
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, a + b = 1 ∧ ‖a • x + b • y‖ < 1) :
StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex' fun x hx y hy hne => ?_)
rw [interior_closedBall (0 : E) one_ne_zero, closedBall_diff_ball,
mem_sphere_zero_iff_norm] at hx hy
rcases h x y hx hy hne with ⟨a, b, hab, hlt⟩
use b
rwa [AffineMap.lineMap_apply_module, interior_closedBall (0 : E) one_ne_zero, mem_ball_zero_iff,
sub_eq_iff_eq_add.2 hab.symm]
#align strict_convex_space.of_norm_combo_lt_one StrictConvexSpace.of_norm_combo_lt_one
theorem StrictConvexSpace.of_norm_combo_ne_one
(h :
∀ x y : E,
‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b : ℝ, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1) :
StrictConvexSpace ℝ E := by
refine StrictConvexSpace.of_strictConvex_closed_unit_ball ℝ
((convex_closedBall _ _).strictConvex ?_)
simp only [interior_closedBall _ one_ne_zero, closedBall_diff_ball, Set.Pairwise,
frontier_closedBall _ one_ne_zero, mem_sphere_zero_iff_norm]
intro x hx y hy hne
rcases h x y hx hy hne with ⟨a, b, ha, hb, hab, hne'⟩
exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩
#align strict_convex_space.of_norm_combo_ne_one StrictConvexSpace.of_norm_combo_ne_one
theorem StrictConvexSpace.of_norm_add_ne_two
(h : ∀ ⦃x y : E⦄, ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E := by
refine
StrictConvexSpace.of_norm_combo_ne_one fun x y hx hy hne =>
⟨1 / 2, 1 / 2, one_half_pos.le, one_half_pos.le, add_halves _, ?_⟩
rw [← smul_add, norm_smul, Real.norm_of_nonneg one_half_pos.le, one_div, ← div_eq_inv_mul, Ne,
div_eq_one_iff_eq (two_ne_zero' ℝ)]
exact h hx hy hne
#align strict_convex_space.of_norm_add_ne_two StrictConvexSpace.of_norm_add_ne_two
theorem StrictConvexSpace.of_pairwise_sphere_norm_ne_two
(h : (sphere (0 : E) 1).Pairwise fun x y => ‖x + y‖ ≠ 2) : StrictConvexSpace ℝ E :=
StrictConvexSpace.of_norm_add_ne_two fun _ _ hx hy =>
h (mem_sphere_zero_iff_norm.2 hx) (mem_sphere_zero_iff_norm.2 hy)
#align strict_convex_space.of_pairwise_sphere_norm_ne_two StrictConvexSpace.of_pairwise_sphere_norm_ne_two
theorem StrictConvexSpace.of_norm_add
(h : ∀ x y : E, ‖x‖ = 1 → ‖y‖ = 1 → ‖x + y‖ = 2 → SameRay ℝ x y) : StrictConvexSpace ℝ E := by
refine StrictConvexSpace.of_pairwise_sphere_norm_ne_two fun x hx y hy => mt fun h₂ => ?_
rw [mem_sphere_zero_iff_norm] at hx hy
exact (sameRay_iff_of_norm_eq (hx.trans hy.symm)).1 (h x y hx hy h₂)
#align strict_convex_space.of_norm_add StrictConvexSpace.of_norm_add
variable [StrictConvexSpace ℝ E] {x y z : E} {a b r : ℝ}
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 152 | 158 | theorem combo_mem_ball_of_ne (hx : x ∈ closedBall z r) (hy : y ∈ closedBall z r) (hne : x ≠ y)
(ha : 0 < a) (hb : 0 < b) (hab : a + b = 1) : a • x + b • y ∈ ball z r := by |
rcases eq_or_ne r 0 with (rfl | hr)
· rw [closedBall_zero, mem_singleton_iff] at hx hy
exact (hne (hx.trans hy.symm)).elim
· simp only [← interior_closedBall _ hr] at hx hy ⊢
exact strictConvex_closedBall ℝ z r hx hy hne ha hb hab
| 2,127 |
import Mathlib.Analysis.Convex.StrictConvexSpace
#align_import analysis.convex.uniform from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open Set Metric
open Convex Pointwise
class UniformConvexSpace (E : Type*) [SeminormedAddCommGroup E] : Prop where
uniform_convex : ∀ ⦃ε : ℝ⦄,
0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
#align uniform_convex_space UniformConvexSpace
variable {E : Type*}
section SeminormedAddCommGroup
variable (E) [SeminormedAddCommGroup E] [UniformConvexSpace E] {ε : ℝ}
theorem exists_forall_sphere_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ :=
UniformConvexSpace.uniform_convex hε
#align exists_forall_sphere_dist_add_le_two_sub exists_forall_sphere_dist_add_le_two_sub
variable [NormedSpace ℝ E]
| Mathlib/Analysis/Convex/Uniform.lean | 60 | 112 | theorem exists_forall_closed_ball_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y⦄, ‖y‖ ≤ 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ := by |
have hε' : 0 < ε / 3 := div_pos hε zero_lt_three
obtain ⟨δ, hδ, h⟩ := exists_forall_sphere_dist_add_le_two_sub E hε'
set δ' := min (1 / 2) (min (ε / 3) <| δ / 3)
refine ⟨δ', lt_min one_half_pos <| lt_min hε' (div_pos hδ zero_lt_three), fun x hx y hy hxy => ?_⟩
obtain hx' | hx' := le_or_lt ‖x‖ (1 - δ')
· rw [← one_add_one_eq_two]
exact (norm_add_le_of_le hx' hy).trans (sub_add_eq_add_sub _ _ _).le
obtain hy' | hy' := le_or_lt ‖y‖ (1 - δ')
· rw [← one_add_one_eq_two]
exact (norm_add_le_of_le hx hy').trans (add_sub_assoc _ _ _).ge
have hδ' : 0 < 1 - δ' := sub_pos_of_lt (min_lt_of_left_lt one_half_lt_one)
have h₁ : ∀ z : E, 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1 := by
rintro z hz
rw [norm_smul_of_nonneg (inv_nonneg.2 <| norm_nonneg _), inv_mul_cancel (hδ'.trans hz).ne']
have h₂ : ∀ z : E, ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ' := by
rintro z hz hδz
nth_rw 3 [← one_smul ℝ z]
rwa [← sub_smul, norm_smul_of_nonneg (sub_nonneg_of_le <| one_le_inv (hδ'.trans_le hδz) hz),
sub_mul, inv_mul_cancel (hδ'.trans_le hδz).ne', one_mul, sub_le_comm]
set x' := ‖x‖⁻¹ • x
set y' := ‖y‖⁻¹ • y
have hxy' : ε / 3 ≤ ‖x' - y'‖ :=
calc
ε / 3 = ε - (ε / 3 + ε / 3) := by ring
_ ≤ ‖x - y‖ - (‖x' - x‖ + ‖y' - y‖) := by
gcongr
· exact (h₂ _ hx hx'.le).trans <| min_le_of_right_le <| min_le_left _ _
· exact (h₂ _ hy hy'.le).trans <| min_le_of_right_le <| min_le_left _ _
_ ≤ _ := by
have : ∀ x' y', x - y = x' - y' + (x - x') + (y' - y) := fun _ _ => by abel
rw [sub_le_iff_le_add, norm_sub_rev _ x, ← add_assoc, this]
exact norm_add₃_le _ _ _
calc
‖x + y‖ ≤ ‖x' + y'‖ + ‖x' - x‖ + ‖y' - y‖ := by
have : ∀ x' y', x + y = x' + y' + (x - x') + (y - y') := fun _ _ => by abel
rw [norm_sub_rev, norm_sub_rev y', this]
exact norm_add₃_le _ _ _
_ ≤ 2 - δ + δ' + δ' :=
(add_le_add_three (h (h₁ _ hx') (h₁ _ hy') hxy') (h₂ _ hx hx'.le) (h₂ _ hy hy'.le))
_ ≤ 2 - δ' := by
dsimp [δ']
rw [← le_sub_iff_add_le, ← le_sub_iff_add_le, sub_sub, sub_sub]
refine sub_le_sub_left ?_ _
ring_nf
rw [← mul_div_cancel₀ δ three_ne_zero]
set_option tactic.skipAssignedInstances false in norm_num
-- Porting note: these three extra lines needed to make `exact` work
have : 3 * (δ / 3) * (1 / 3) = δ / 3 := by linarith
rw [this, mul_comm]
gcongr
exact min_le_of_right_le <| min_le_right _ _
| 2,128 |
import Mathlib.Analysis.Convex.StrictConvexSpace
#align_import analysis.convex.uniform from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open Set Metric
open Convex Pointwise
class UniformConvexSpace (E : Type*) [SeminormedAddCommGroup E] : Prop where
uniform_convex : ∀ ⦃ε : ℝ⦄,
0 < ε → ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ
#align uniform_convex_space UniformConvexSpace
variable {E : Type*}
section SeminormedAddCommGroup
variable (E) [SeminormedAddCommGroup E] [UniformConvexSpace E] {ε : ℝ}
theorem exists_forall_sphere_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ = 1 → ∀ ⦃y⦄, ‖y‖ = 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ :=
UniformConvexSpace.uniform_convex hε
#align exists_forall_sphere_dist_add_le_two_sub exists_forall_sphere_dist_add_le_two_sub
variable [NormedSpace ℝ E]
theorem exists_forall_closed_ball_dist_add_le_two_sub (hε : 0 < ε) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ 1 → ∀ ⦃y⦄, ‖y‖ ≤ 1 → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 - δ := by
have hε' : 0 < ε / 3 := div_pos hε zero_lt_three
obtain ⟨δ, hδ, h⟩ := exists_forall_sphere_dist_add_le_two_sub E hε'
set δ' := min (1 / 2) (min (ε / 3) <| δ / 3)
refine ⟨δ', lt_min one_half_pos <| lt_min hε' (div_pos hδ zero_lt_three), fun x hx y hy hxy => ?_⟩
obtain hx' | hx' := le_or_lt ‖x‖ (1 - δ')
· rw [← one_add_one_eq_two]
exact (norm_add_le_of_le hx' hy).trans (sub_add_eq_add_sub _ _ _).le
obtain hy' | hy' := le_or_lt ‖y‖ (1 - δ')
· rw [← one_add_one_eq_two]
exact (norm_add_le_of_le hx hy').trans (add_sub_assoc _ _ _).ge
have hδ' : 0 < 1 - δ' := sub_pos_of_lt (min_lt_of_left_lt one_half_lt_one)
have h₁ : ∀ z : E, 1 - δ' < ‖z‖ → ‖‖z‖⁻¹ • z‖ = 1 := by
rintro z hz
rw [norm_smul_of_nonneg (inv_nonneg.2 <| norm_nonneg _), inv_mul_cancel (hδ'.trans hz).ne']
have h₂ : ∀ z : E, ‖z‖ ≤ 1 → 1 - δ' ≤ ‖z‖ → ‖‖z‖⁻¹ • z - z‖ ≤ δ' := by
rintro z hz hδz
nth_rw 3 [← one_smul ℝ z]
rwa [← sub_smul, norm_smul_of_nonneg (sub_nonneg_of_le <| one_le_inv (hδ'.trans_le hδz) hz),
sub_mul, inv_mul_cancel (hδ'.trans_le hδz).ne', one_mul, sub_le_comm]
set x' := ‖x‖⁻¹ • x
set y' := ‖y‖⁻¹ • y
have hxy' : ε / 3 ≤ ‖x' - y'‖ :=
calc
ε / 3 = ε - (ε / 3 + ε / 3) := by ring
_ ≤ ‖x - y‖ - (‖x' - x‖ + ‖y' - y‖) := by
gcongr
· exact (h₂ _ hx hx'.le).trans <| min_le_of_right_le <| min_le_left _ _
· exact (h₂ _ hy hy'.le).trans <| min_le_of_right_le <| min_le_left _ _
_ ≤ _ := by
have : ∀ x' y', x - y = x' - y' + (x - x') + (y' - y) := fun _ _ => by abel
rw [sub_le_iff_le_add, norm_sub_rev _ x, ← add_assoc, this]
exact norm_add₃_le _ _ _
calc
‖x + y‖ ≤ ‖x' + y'‖ + ‖x' - x‖ + ‖y' - y‖ := by
have : ∀ x' y', x + y = x' + y' + (x - x') + (y - y') := fun _ _ => by abel
rw [norm_sub_rev, norm_sub_rev y', this]
exact norm_add₃_le _ _ _
_ ≤ 2 - δ + δ' + δ' :=
(add_le_add_three (h (h₁ _ hx') (h₁ _ hy') hxy') (h₂ _ hx hx'.le) (h₂ _ hy hy'.le))
_ ≤ 2 - δ' := by
dsimp [δ']
rw [← le_sub_iff_add_le, ← le_sub_iff_add_le, sub_sub, sub_sub]
refine sub_le_sub_left ?_ _
ring_nf
rw [← mul_div_cancel₀ δ three_ne_zero]
set_option tactic.skipAssignedInstances false in norm_num
-- Porting note: these three extra lines needed to make `exact` work
have : 3 * (δ / 3) * (1 / 3) = δ / 3 := by linarith
rw [this, mul_comm]
gcongr
exact min_le_of_right_le <| min_le_right _ _
#align exists_forall_closed_ball_dist_add_le_two_sub exists_forall_closed_ball_dist_add_le_two_sub
| Mathlib/Analysis/Convex/Uniform.lean | 115 | 126 | theorem exists_forall_closed_ball_dist_add_le_two_mul_sub (hε : 0 < ε) (r : ℝ) :
∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ := by |
obtain hr | hr := le_or_lt r 0
· exact ⟨1, one_pos, fun x hx y hy h => (hε.not_le <|
h.trans <| (norm_sub_le _ _).trans <| add_nonpos (hx.trans hr) (hy.trans hr)).elim⟩
obtain ⟨δ, hδ, h⟩ := exists_forall_closed_ball_dist_add_le_two_sub E (div_pos hε hr)
refine ⟨δ * r, mul_pos hδ hr, fun x hx y hy hxy => ?_⟩
rw [← div_le_one hr, div_eq_inv_mul, ← norm_smul_of_nonneg (inv_nonneg.2 hr.le)] at hx hy
have := h hx hy
simp_rw [← smul_add, ← smul_sub, norm_smul_of_nonneg (inv_nonneg.2 hr.le), ← div_eq_inv_mul,
div_le_div_right hr, div_le_iff hr, sub_mul] at this
exact this hxy
| 2,128 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
| Mathlib/Analysis/Convex/Side.lean | 62 | 67 | theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by |
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
| 2,129 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_same_side.map AffineSubspace.WSameSide.map
| Mathlib/Analysis/Convex/Side.lean | 70 | 80 | theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
| 2,129 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_same_side.map AffineSubspace.WSameSide.map
theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
#align function.injective.w_same_side_map_iff Function.Injective.wSameSide_map_iff
| Mathlib/Analysis/Convex/Side.lean | 83 | 86 | theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y := by |
simp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]
| 2,129 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_same_side.map AffineSubspace.WSameSide.map
theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
#align function.injective.w_same_side_map_iff Function.Injective.wSameSide_map_iff
theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y := by
simp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]
#align function.injective.s_same_side_map_iff Function.Injective.sSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).WSameSide (f x) (f y) ↔ s.WSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).wSameSide_map_iff
#align affine_equiv.w_same_side_map_iff AffineEquiv.wSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).SSameSide (f x) (f y) ↔ s.SSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).sSameSide_map_iff
#align affine_equiv.s_same_side_map_iff AffineEquiv.sSameSide_map_iff
| Mathlib/Analysis/Convex/Side.lean | 101 | 106 | theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
(s.map f).WOppSide (f x) (f y) := by |
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
| 2,129 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace AffineSubspace
section StrictOrderedCommRing
variable [StrictOrderedCommRing R] [AddCommGroup V] [Module R V] [AddTorsor V P]
variable [AddCommGroup V'] [Module R V'] [AddTorsor V' P']
def WSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (y -ᵥ p₂)
#align affine_subspace.w_same_side AffineSubspace.WSameSide
def SSameSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WSameSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_same_side AffineSubspace.SSameSide
def WOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
∃ᵉ (p₁ ∈ s) (p₂ ∈ s), SameRay R (x -ᵥ p₁) (p₂ -ᵥ y)
#align affine_subspace.w_opp_side AffineSubspace.WOppSide
def SOppSide (s : AffineSubspace R P) (x y : P) : Prop :=
s.WOppSide x y ∧ x ∉ s ∧ y ∉ s
#align affine_subspace.s_opp_side AffineSubspace.SOppSide
theorem WSameSide.map {s : AffineSubspace R P} {x y : P} (h : s.WSameSide x y) (f : P →ᵃ[R] P') :
(s.map f).WSameSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_same_side.map AffineSubspace.WSameSide.map
theorem _root_.Function.Injective.wSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WSameSide (f x) (f y) ↔ s.WSameSide x y := by
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
#align function.injective.w_same_side_map_iff Function.Injective.wSameSide_map_iff
theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y := by
simp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]
#align function.injective.s_same_side_map_iff Function.Injective.sSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.wSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).WSameSide (f x) (f y) ↔ s.WSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).wSameSide_map_iff
#align affine_equiv.w_same_side_map_iff AffineEquiv.wSameSide_map_iff
@[simp]
theorem _root_.AffineEquiv.sSameSide_map_iff {s : AffineSubspace R P} {x y : P} (f : P ≃ᵃ[R] P') :
(s.map ↑f).SSameSide (f x) (f y) ↔ s.SSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).sSameSide_map_iff
#align affine_equiv.s_same_side_map_iff AffineEquiv.sSameSide_map_iff
theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
(s.map f).WOppSide (f x) (f y) := by
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
#align affine_subspace.w_opp_side.map AffineSubspace.WOppSide.map
| Mathlib/Analysis/Convex/Side.lean | 109 | 119 | theorem _root_.Function.Injective.wOppSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).WOppSide (f x) (f y) ↔ s.WOppSide x y := by |
refine ⟨fun h => ?_, fun h => h.map _⟩
rcases h with ⟨fp₁, hfp₁, fp₂, hfp₂, h⟩
rw [mem_map] at hfp₁ hfp₂
rcases hfp₁ with ⟨p₁, hp₁, rfl⟩
rcases hfp₂ with ⟨p₂, hp₂, rfl⟩
refine ⟨p₁, hp₁, p₂, hp₂, ?_⟩
simp_rw [← linearMap_vsub, (f.linear_injective_iff.2 hf).sameRay_map_iff] at h
exact h
| 2,129 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
| Mathlib/Analysis/Convex/GaugeRescale.lean | 41 | 44 | theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by |
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
| Mathlib/Analysis/Convex/GaugeRescale.lean | 48 | 52 | theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by |
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
gaugeRescale s s = id :=
funext <| gaugeRescale_self_apply hsa hsb
| Mathlib/Analysis/Convex/GaugeRescale.lean | 58 | 61 | theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by |
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
gaugeRescale s s = id :=
funext <| gaugeRescale_self_apply hsa hsb
theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
| Mathlib/Analysis/Convex/GaugeRescale.lean | 63 | 67 | theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by |
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
gaugeRescale s s = id :=
funext <| gaugeRescale_self_apply hsa hsb
theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
| Mathlib/Analysis/Convex/GaugeRescale.lean | 69 | 73 | theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by |
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
gaugeRescale s s = id :=
funext <| gaugeRescale_self_apply hsa hsb
theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
| Mathlib/Analysis/Convex/GaugeRescale.lean | 75 | 80 | theorem gaugeRescale_gaugeRescale {s t u : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x := by |
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale_def s t x, gaugeRescale_smul, gaugeRescale, gaugeRescale, smul_smul,
div_mul_div_cancel]
exacts [((gauge_pos hta htb).2 hx).ne', div_nonneg (gauge_nonneg _) (gauge_nonneg _)]
| 2,130 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
theorem gaugeRescale_def (s t : Set E) (x : E) :
gaugeRescale s t x = (gauge s x / gauge t x) • x :=
rfl
@[simp] theorem gaugeRescale_zero (s t : Set E) : gaugeRescale s t 0 = 0 := smul_zero _
theorem gaugeRescale_smul (s t : Set E) {c : ℝ} (hc : 0 ≤ c) (x : E) :
gaugeRescale s t (c • x) = c • gaugeRescale s t x := by
simp only [gaugeRescale, gauge_smul_of_nonneg hc, smul_smul, smul_eq_mul]
rw [mul_div_mul_comm, mul_right_comm, div_self_mul_self]
variable [TopologicalSpace E] [T1Space E]
theorem gaugeRescale_self_apply {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(x : E) : gaugeRescale s s x = x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale, div_self, one_smul]
exact ((gauge_pos hsa hsb).2 hx).ne'
theorem gaugeRescale_self {s : Set E} (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s) :
gaugeRescale s s = id :=
funext <| gaugeRescale_self_apply hsa hsb
theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
theorem gauge_gaugeRescale (s : Set E) {t : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gauge t (gaugeRescale s t x) = gauge s x := by
rcases eq_or_ne x 0 with rfl | hx
· simp
· exact gauge_gaugeRescale' s ((gauge_pos hta htb).2 hx).ne'
theorem gauge_gaugeRescale_le (s t : Set E) (x : E) :
gauge t (gaugeRescale s t x) ≤ gauge s x := by
by_cases hx : gauge t x = 0
· simp [gaugeRescale, hx, gauge_nonneg]
· exact (gauge_gaugeRescale' s hx).le
theorem gaugeRescale_gaugeRescale {s t u : Set E} (hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t)
(x : E) : gaugeRescale t u (gaugeRescale s t x) = gaugeRescale s u x := by
rcases eq_or_ne x 0 with rfl | hx; · simp
rw [gaugeRescale_def s t x, gaugeRescale_smul, gaugeRescale, gaugeRescale, smul_smul,
div_mul_div_cancel]
exacts [((gauge_pos hta htb).2 hx).ne', div_nonneg (gauge_nonneg _) (gauge_nonneg _)]
def gaugeRescaleEquiv (s t : Set E) (hsa : Absorbent ℝ s) (hsb : IsVonNBounded ℝ s)
(hta : Absorbent ℝ t) (htb : IsVonNBounded ℝ t) : E ≃ E where
toFun := gaugeRescale s t
invFun := gaugeRescale t s
left_inv x := by rw [gaugeRescale_gaugeRescale, gaugeRescale_self_apply] <;> assumption
right_inv x := by rw [gaugeRescale_gaugeRescale, gaugeRescale_self_apply] <;> assumption
variable [TopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E}
theorem mapsTo_gaugeRescale_interior (h₀ : t ∈ 𝓝 0) (hc : Convex ℝ t) :
MapsTo (gaugeRescale s t) (interior s) (interior t) := fun x hx ↦ by
rw [← gauge_lt_one_iff_mem_interior] <;> try assumption
exact (gauge_gaugeRescale_le _ _ _).trans_lt (interior_subset_gauge_lt_one _ hx)
theorem mapsTo_gaugeRescale_closure {s t : Set E} (hsc : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
(htc : Convex ℝ t) (ht₀ : 0 ∈ t) (hta : Absorbent ℝ t) :
MapsTo (gaugeRescale s t) (closure s) (closure t) := fun _x hx ↦
mem_closure_of_gauge_le_one htc ht₀ hta <| (gauge_gaugeRescale_le _ _ _).trans <|
(gauge_le_one_iff_mem_closure hsc hs₀).2 hx
| Mathlib/Analysis/Convex/GaugeRescale.lean | 103 | 114 | theorem continuous_gaugeRescale {s t : Set E} (hs : Convex ℝ s) (hs₀ : s ∈ 𝓝 0)
(ht : Convex ℝ t) (ht₀ : t ∈ 𝓝 0) (htb : IsVonNBounded ℝ t) :
Continuous (gaugeRescale s t) := by |
have hta : Absorbent ℝ t := absorbent_nhds_zero ht₀
refine continuous_iff_continuousAt.2 fun x ↦ ?_
rcases eq_or_ne x 0 with rfl | hx
· rw [ContinuousAt, gaugeRescale_zero]
nth_rewrite 2 [← comap_gauge_nhds_zero htb ht₀]
simp only [tendsto_comap_iff, (· ∘ ·), gauge_gaugeRescale _ hta htb]
exact tendsto_gauge_nhds_zero hs₀
· exact ((continuousAt_gauge hs hs₀).div (continuousAt_gauge ht ht₀)
((gauge_pos hta htb).2 hx).ne').smul continuousAt_id
| 2,130 |
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics ContinuousLinearMap Filter
open scoped Classical Topology NNReal
| Mathlib/Analysis/Calculus/MeanValue.lean | 92 | 124 | theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by |
change Icc a b ⊆ { x | f x ≤ B x }
set s := { x | f x ≤ B x } ∩ Icc a b
have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
have : IsClosed s := by
simp only [s, inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
apply this.Icc_subset_of_forall_exists_gt ha
rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
cases' hxB.lt_or_eq with hxB hxB
· -- If `f x < B x`, then all we need is continuity of both sides
refine nonempty_of_mem (inter_mem ?_ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
exact this.mono fun y => le_of_lt
· rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
specialize hf' x xab r hfr
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB)
obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y :=
(hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
refine ⟨z, ?_, hz⟩
have := (hfz.trans hzB).le
rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
sub_le_sub_iff_right] at this
| 2,131 |
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.LocalExtr.Rolle
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.calculus.mean_value from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics ContinuousLinearMap Filter
open scoped Classical Topology NNReal
theorem image_le_of_liminf_slope_right_lt_deriv_boundary' {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by
change Icc a b ⊆ { x | f x ≤ B x }
set s := { x | f x ≤ B x } ∩ Icc a b
have A : ContinuousOn (fun x => (f x, B x)) (Icc a b) := hf.prod hB
have : IsClosed s := by
simp only [s, inter_comm]
exact A.preimage_isClosed_of_isClosed isClosed_Icc OrderClosedTopology.isClosed_le'
apply this.Icc_subset_of_forall_exists_gt ha
rintro x ⟨hxB : f x ≤ B x, xab⟩ y hy
cases' hxB.lt_or_eq with hxB hxB
· -- If `f x < B x`, then all we need is continuity of both sides
refine nonempty_of_mem (inter_mem ?_ (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))
have : ∀ᶠ x in 𝓝[Icc a b] x, f x < B x :=
A x (Ico_subset_Icc_self xab) (IsOpen.mem_nhds (isOpen_lt continuous_fst continuous_snd) hxB)
have : ∀ᶠ x in 𝓝[>] x, f x < B x := nhdsWithin_le_of_mem (Icc_mem_nhdsWithin_Ioi xab) this
exact this.mono fun y => le_of_lt
· rcases exists_between (bound x xab hxB) with ⟨r, hfr, hrB⟩
specialize hf' x xab r hfr
have HB : ∀ᶠ z in 𝓝[>] x, r < slope B x z :=
(hasDerivWithinAt_iff_tendsto_slope' <| lt_irrefl x).1 (hB' x xab).Ioi_of_Ici
(Ioi_mem_nhds hrB)
obtain ⟨z, hfz, hzB, hz⟩ : ∃ z, slope f x z < r ∧ r < slope B x z ∧ z ∈ Ioc x y :=
(hf'.and_eventually (HB.and (Ioc_mem_nhdsWithin_Ioi ⟨le_rfl, hy⟩))).exists
refine ⟨z, ?_, hz⟩
have := (hfz.trans hzB).le
rwa [slope_def_field, slope_def_field, div_le_div_right (sub_pos.2 hz.1), hxB,
sub_le_sub_iff_right] at this
#align image_le_of_liminf_slope_right_lt_deriv_boundary' image_le_of_liminf_slope_right_lt_deriv_boundary'
theorem image_le_of_liminf_slope_right_lt_deriv_boundary {f f' : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b))
-- `hf'` actually says `liminf (f z - f x) / (z - x) ≤ f' x`
(hf' : ∀ x ∈ Ico a b, ∀ r, f' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r)
{B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ∀ x, HasDerivAt B (B' x) x)
(bound : ∀ x ∈ Ico a b, f x = B x → f' x < B' x) : ∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x :=
image_le_of_liminf_slope_right_lt_deriv_boundary' hf hf' ha
(fun x _ => (hB x).continuousAt.continuousWithinAt) (fun x _ => (hB x).hasDerivWithinAt) bound
#align image_le_of_liminf_slope_right_lt_deriv_boundary image_le_of_liminf_slope_right_lt_deriv_boundary
| Mathlib/Analysis/Calculus/MeanValue.lean | 156 | 175 | theorem image_le_of_liminf_slope_right_le_deriv_boundary {f : ℝ → ℝ} {a b : ℝ}
(hf : ContinuousOn f (Icc a b)) {B B' : ℝ → ℝ} (ha : f a ≤ B a) (hB : ContinuousOn B (Icc a b))
(hB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x)
-- `bound` actually says `liminf (f z - f x) / (z - x) ≤ B' x`
(bound : ∀ x ∈ Ico a b, ∀ r, B' x < r → ∃ᶠ z in 𝓝[>] x, slope f x z < r) :
∀ ⦃x⦄, x ∈ Icc a b → f x ≤ B x := by |
have Hr : ∀ x ∈ Icc a b, ∀ r > 0, f x ≤ B x + r * (x - a) := fun x hx r hr => by
apply image_le_of_liminf_slope_right_lt_deriv_boundary' hf bound
· rwa [sub_self, mul_zero, add_zero]
· exact hB.add (continuousOn_const.mul (continuousOn_id.sub continuousOn_const))
· intro x hx
exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r)
· intro x _ _
rw [mul_one]
exact (lt_add_iff_pos_right _).2 hr
exact hx
intro x hx
have : ContinuousWithinAt (fun r => B x + r * (x - a)) (Ioi 0) 0 :=
continuousWithinAt_const.add (continuousWithinAt_id.mul continuousWithinAt_const)
convert continuousWithinAt_const.closure_le _ this (Hr x hx) using 1 <;> simp
| 2,131 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ}
namespace HasDerivAt
| Mathlib/Analysis/Calculus/LHopital.lean | 51 | 92 | theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by |
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx =>
Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)
have hg : ∀ x ∈ Ioo a b, g x ≠ 0 := by
intro x hx h
have : Tendsto g (𝓝[<] x) (𝓝 0) := by
rw [← h, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]
exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto
obtain ⟨y, hyx, hy⟩ : ∃ c ∈ Ioo a x, g' c = 0 :=
exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy
exact hg' y (sub x hx hyx) hy
have : ∀ x ∈ Ioo a b, ∃ c ∈ Ioo a x, f x * g' c = g x * f' c := by
intro x hx
rw [← sub_zero (f x), ← sub_zero (g x)]
exact exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)
(fun y hy => hff' y <| sub x hx hy) hga hfa
(tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto)
(tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto)
choose! c hc using this
have : ∀ x ∈ Ioo a b, ((fun x' => f' x' / g' x') ∘ c) x = f x / g x := by
intro x hx
rcases hc x hx with ⟨h₁, h₂⟩
field_simp [hg x hx, hg' (c x) ((sub x hx) h₁)]
simp only [h₂]
rw [mul_comm]
have cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x := fun x hx => (hc x hx).1
rw [← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
apply tendsto_nhdsWithin_congr this
apply hdiv.comp
refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
(tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds
(tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) ?_ ?_) ?_
all_goals
apply eventually_nhdsWithin_of_forall
intro x hx
have := cmp x hx
try simp
linarith [this]
| 2,132 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ}
namespace HasDerivAt
theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx =>
Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)
have hg : ∀ x ∈ Ioo a b, g x ≠ 0 := by
intro x hx h
have : Tendsto g (𝓝[<] x) (𝓝 0) := by
rw [← h, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]
exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto
obtain ⟨y, hyx, hy⟩ : ∃ c ∈ Ioo a x, g' c = 0 :=
exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy
exact hg' y (sub x hx hyx) hy
have : ∀ x ∈ Ioo a b, ∃ c ∈ Ioo a x, f x * g' c = g x * f' c := by
intro x hx
rw [← sub_zero (f x), ← sub_zero (g x)]
exact exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)
(fun y hy => hff' y <| sub x hx hy) hga hfa
(tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto)
(tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto)
choose! c hc using this
have : ∀ x ∈ Ioo a b, ((fun x' => f' x' / g' x') ∘ c) x = f x / g x := by
intro x hx
rcases hc x hx with ⟨h₁, h₂⟩
field_simp [hg x hx, hg' (c x) ((sub x hx) h₁)]
simp only [h₂]
rw [mul_comm]
have cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x := fun x hx => (hc x hx).1
rw [← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
apply tendsto_nhdsWithin_congr this
apply hdiv.comp
refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
(tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds
(tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) ?_ ?_) ?_
all_goals
apply eventually_nhdsWithin_of_forall
intro x hx
have := cmp x hx
try simp
linarith [this]
#align has_deriv_at.lhopital_zero_right_on_Ioo HasDerivAt.lhopital_zero_right_on_Ioo
| Mathlib/Analysis/Calculus/LHopital.lean | 95 | 104 | theorem lhopital_zero_right_on_Ico (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b))
(hcg : ContinuousOn g (Ico a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by |
refine lhopital_zero_right_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv
· rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto
· rw [← hga, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto
| 2,132 |
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Set
open scoped Filter Topology Pointwise
variable {a b : ℝ} (hab : a < b) {l : Filter ℝ} {f f' g g' : ℝ → ℝ}
namespace HasDerivAt
theorem lhopital_zero_right_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfa : Tendsto f (𝓝[>] a) (𝓝 0)) (hga : Tendsto g (𝓝[>] a) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by
have sub : ∀ x ∈ Ioo a b, Ioo a x ⊆ Ioo a b := fun x hx =>
Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)
have hg : ∀ x ∈ Ioo a b, g x ≠ 0 := by
intro x hx h
have : Tendsto g (𝓝[<] x) (𝓝 0) := by
rw [← h, ← nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]
exact ((hgg' x hx).continuousAt.continuousWithinAt.mono <| sub x hx).tendsto
obtain ⟨y, hyx, hy⟩ : ∃ c ∈ Ioo a x, g' c = 0 :=
exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy
exact hg' y (sub x hx hyx) hy
have : ∀ x ∈ Ioo a b, ∃ c ∈ Ioo a x, f x * g' c = g x * f' c := by
intro x hx
rw [← sub_zero (f x), ← sub_zero (g x)]
exact exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)
(fun y hy => hff' y <| sub x hx hy) hga hfa
(tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).continuousAt.tendsto)
(tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).continuousAt.tendsto)
choose! c hc using this
have : ∀ x ∈ Ioo a b, ((fun x' => f' x' / g' x') ∘ c) x = f x / g x := by
intro x hx
rcases hc x hx with ⟨h₁, h₂⟩
field_simp [hg x hx, hg' (c x) ((sub x hx) h₁)]
simp only [h₂]
rw [mul_comm]
have cmp : ∀ x ∈ Ioo a b, a < c x ∧ c x < x := fun x hx => (hc x hx).1
rw [← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
apply tendsto_nhdsWithin_congr this
apply hdiv.comp
refine tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _
(tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds
(tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) ?_ ?_) ?_
all_goals
apply eventually_nhdsWithin_of_forall
intro x hx
have := cmp x hx
try simp
linarith [this]
#align has_deriv_at.lhopital_zero_right_on_Ioo HasDerivAt.lhopital_zero_right_on_Ioo
theorem lhopital_zero_right_on_Ico (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b))
(hcg : ContinuousOn g (Ico a b)) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0) (hfa : f a = 0) (hga : g a = 0)
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[>] a) l) :
Tendsto (fun x => f x / g x) (𝓝[>] a) l := by
refine lhopital_zero_right_on_Ioo hab hff' hgg' hg' ?_ ?_ hdiv
· rw [← hfa, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto
· rw [← hga, ← nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).tendsto
#align has_deriv_at.lhopital_zero_right_on_Ico HasDerivAt.lhopital_zero_right_on_Ico
| Mathlib/Analysis/Calculus/LHopital.lean | 107 | 129 | theorem lhopital_zero_left_on_Ioo (hff' : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x)
(hgg' : ∀ x ∈ Ioo a b, HasDerivAt g (g' x) x) (hg' : ∀ x ∈ Ioo a b, g' x ≠ 0)
(hfb : Tendsto f (𝓝[<] b) (𝓝 0)) (hgb : Tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : Tendsto (fun x => f' x / g' x) (𝓝[<] b) l) :
Tendsto (fun x => f x / g x) (𝓝[<] b) l := by |
-- Here, we essentially compose by `Neg.neg`. The following is mostly technical details.
have hdnf : ∀ x ∈ -Ioo a b, HasDerivAt (f ∘ Neg.neg) (f' (-x) * -1) x := fun x hx =>
comp x (hff' (-x) hx) (hasDerivAt_neg x)
have hdng : ∀ x ∈ -Ioo a b, HasDerivAt (g ∘ Neg.neg) (g' (-x) * -1) x := fun x hx =>
comp x (hgg' (-x) hx) (hasDerivAt_neg x)
rw [preimage_neg_Ioo] at hdnf
rw [preimage_neg_Ioo] at hdng
have := lhopital_zero_right_on_Ioo (neg_lt_neg hab) hdnf hdng (by
intro x hx h
apply hg' _ (by rw [← preimage_neg_Ioo] at hx; exact hx)
rwa [mul_comm, ← neg_eq_neg_one_mul, neg_eq_zero] at h)
(hfb.comp tendsto_neg_nhdsWithin_Ioi_neg) (hgb.comp tendsto_neg_nhdsWithin_Ioi_neg)
(by
simp only [neg_div_neg_eq, mul_one, mul_neg]
exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_nhdsWithin_Ioi_neg))
have := this.comp tendsto_neg_nhdsWithin_Iio
unfold Function.comp at this
simpa only [neg_neg]
| 2,132 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variable {n : ℕ∞} {𝕂 : Type*} [RCLike 𝕂] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
{F' : Type*} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
theorem HasFTaylorSeriesUpToOn.hasStrictFDerivAt {s : Set E'} {f : E' → F'} {x : E'}
{p : E' → FormalMultilinearSeries 𝕂 E' F'} (hf : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
(hs : s ∈ 𝓝 x) : HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x :=
hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt (hf.eventually_hasFDerivAt hn hs) <|
(continuousMultilinearCurryFin1 𝕂 E' F').continuousAt.comp <| (hf.cont 1 hn).continuousAt hs
#align has_ftaylor_series_up_to_on.has_strict_fderiv_at HasFTaylorSeriesUpToOn.hasStrictFDerivAt
| Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 43 | 49 | theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
HasStrictFDerivAt f f' x := by |
rcases hf 1 hn with ⟨u, H, p, hp⟩
simp only [nhdsWithin_univ, mem_univ, insert_eq_of_mem] at H
have := hp.hasStrictFDerivAt le_rfl H
rwa [hf'.unique this.hasFDerivAt]
| 2,133 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variable {n : ℕ∞} {𝕂 : Type*} [RCLike 𝕂] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕂 E']
{F' : Type*} [NormedAddCommGroup F'] [NormedSpace 𝕂 F']
theorem HasFTaylorSeriesUpToOn.hasStrictFDerivAt {s : Set E'} {f : E' → F'} {x : E'}
{p : E' → FormalMultilinearSeries 𝕂 E' F'} (hf : HasFTaylorSeriesUpToOn n f p s) (hn : 1 ≤ n)
(hs : s ∈ 𝓝 x) : HasStrictFDerivAt f ((continuousMultilinearCurryFin1 𝕂 E' F') (p x 1)) x :=
hasStrictFDerivAt_of_hasFDerivAt_of_continuousAt (hf.eventually_hasFDerivAt hn hs) <|
(continuousMultilinearCurryFin1 𝕂 E' F').continuousAt.comp <| (hf.cont 1 hn).continuousAt hs
#align has_ftaylor_series_up_to_on.has_strict_fderiv_at HasFTaylorSeriesUpToOn.hasStrictFDerivAt
theorem ContDiffAt.hasStrictFDerivAt' {f : E' → F'} {f' : E' →L[𝕂] F'} {x : E'}
(hf : ContDiffAt 𝕂 n f x) (hf' : HasFDerivAt f f' x) (hn : 1 ≤ n) :
HasStrictFDerivAt f f' x := by
rcases hf 1 hn with ⟨u, H, p, hp⟩
simp only [nhdsWithin_univ, mem_univ, insert_eq_of_mem] at H
have := hp.hasStrictFDerivAt le_rfl H
rwa [hf'.unique this.hasFDerivAt]
#align cont_diff_at.has_strict_fderiv_at' ContDiffAt.hasStrictFDerivAt'
theorem ContDiffAt.hasStrictDerivAt' {f : 𝕂 → F'} {f' : F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x)
(hf' : HasDerivAt f f' x) (hn : 1 ≤ n) : HasStrictDerivAt f f' x :=
hf.hasStrictFDerivAt' hf' hn
#align cont_diff_at.has_strict_deriv_at' ContDiffAt.hasStrictDerivAt'
theorem ContDiffAt.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
HasStrictFDerivAt f (fderiv 𝕂 f x) x :=
hf.hasStrictFDerivAt' (hf.differentiableAt hn).hasFDerivAt hn
#align cont_diff_at.has_strict_fderiv_at ContDiffAt.hasStrictFDerivAt
theorem ContDiffAt.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiffAt 𝕂 n f x) (hn : 1 ≤ n) :
HasStrictDerivAt f (deriv f x) x :=
(hf.hasStrictFDerivAt hn).hasStrictDerivAt
#align cont_diff_at.has_strict_deriv_at ContDiffAt.hasStrictDerivAt
theorem ContDiff.hasStrictFDerivAt {f : E' → F'} {x : E'} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
HasStrictFDerivAt f (fderiv 𝕂 f x) x :=
hf.contDiffAt.hasStrictFDerivAt hn
#align cont_diff.has_strict_fderiv_at ContDiff.hasStrictFDerivAt
theorem ContDiff.hasStrictDerivAt {f : 𝕂 → F'} {x : 𝕂} (hf : ContDiff 𝕂 n f) (hn : 1 ≤ n) :
HasStrictDerivAt f (deriv f x) x :=
hf.contDiffAt.hasStrictDerivAt hn
#align cont_diff.has_strict_deriv_at ContDiff.hasStrictDerivAt
| Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 87 | 101 | theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
{p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
(hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) (K : ℝ≥0)
(hK : ‖p x 1‖₊ < K) : ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t := by |
set f' := fun y => continuousMultilinearCurryFin1 ℝ E F (p y 1)
have hder : ∀ y ∈ s, HasFDerivWithinAt f (f' y) s y := fun y hy =>
(hf.hasFDerivWithinAt le_rfl (subset_insert x s hy)).mono (subset_insert x s)
have hcont : ContinuousWithinAt f' s x :=
(continuousMultilinearCurryFin1 ℝ E F).continuousAt.comp_continuousWithinAt
((hf.cont _ le_rfl _ (mem_insert _ _)).mono (subset_insert x s))
replace hK : ‖f' x‖₊ < K := by simpa only [f', LinearIsometryEquiv.nnnorm_map]
exact
hs.exists_nhdsWithin_lipschitzOnWith_of_hasFDerivWithinAt_of_nnnorm_lt
(eventually_nhdsWithin_iff.2 <| eventually_of_forall hder) hcont K hK
| 2,133 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
noncomputable section
namespace ContDiffAt
variable {𝕂 : Type*} [RCLike 𝕂]
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕂 E]
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕂 F]
variable [CompleteSpace E] (f : E → F) {f' : E ≃L[𝕂] F} {a : E}
def toPartialHomeomorph {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a)
(hn : 1 ≤ n) : PartialHomeomorph E F :=
(hf.hasStrictFDerivAt' hf' hn).toPartialHomeomorph f
#align cont_diff_at.to_local_homeomorph ContDiffAt.toPartialHomeomorph
variable {f}
@[simp]
theorem toPartialHomeomorph_coe {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
(hf.toPartialHomeomorph f hf' hn : E → F) = f :=
rfl
#align cont_diff_at.to_local_homeomorph_coe ContDiffAt.toPartialHomeomorph_coe
theorem mem_toPartialHomeomorph_source {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
a ∈ (hf.toPartialHomeomorph f hf' hn).source :=
(hf.hasStrictFDerivAt' hf' hn).mem_toPartialHomeomorph_source
#align cont_diff_at.mem_to_local_homeomorph_source ContDiffAt.mem_toPartialHomeomorph_source
theorem image_mem_toPartialHomeomorph_target {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
f a ∈ (hf.toPartialHomeomorph f hf' hn).target :=
(hf.hasStrictFDerivAt' hf' hn).image_mem_toPartialHomeomorph_target
#align cont_diff_at.image_mem_to_local_homeomorph_target ContDiffAt.image_mem_toPartialHomeomorph_target
def localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (f' : E →L[𝕂] F) a)
(hn : 1 ≤ n) : F → E :=
(hf.hasStrictFDerivAt' hf' hn).localInverse f f' a
#align cont_diff_at.local_inverse ContDiffAt.localInverse
theorem localInverse_apply_image {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) : hf.localInverse hf' hn (f a) = a :=
(hf.hasStrictFDerivAt' hf' hn).localInverse_apply_image
#align cont_diff_at.local_inverse_apply_image ContDiffAt.localInverse_apply_image
| Mathlib/Analysis/Calculus/InverseFunctionTheorem/ContDiff.lean | 68 | 75 | theorem to_localInverse {n : ℕ∞} (hf : ContDiffAt 𝕂 n f a)
(hf' : HasFDerivAt f (f' : E →L[𝕂] F) a) (hn : 1 ≤ n) :
ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a) := by |
have := hf.localInverse_apply_image hf' hn
apply (hf.toPartialHomeomorph f hf' hn).contDiffAt_symm
(image_mem_toPartialHomeomorph_target hf hf' hn)
· convert hf'
· convert hf
| 2,134 |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asymptotics Set Function
open scoped Classical Topology
namespace Complex
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ]
| Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 36 | 42 | theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x := by |
rw [hasDerivAt_iff_isLittleO_nhds_zero]
have : (1 : ℕ) < 2 := by norm_num
refine (IsBigO.of_bound ‖exp x‖ ?_).trans_isLittleO (isLittleO_pow_id this)
filter_upwards [Metric.ball_mem_nhds (0 : ℂ) zero_lt_one]
simp only [Metric.mem_ball, dist_zero_right, norm_pow]
exact fun z hz => exp_bound_sq x z hz.le
| 2,135 |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asymptotics Set Function
open scoped Classical Topology
namespace Complex
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [NormedAlgebra 𝕜 ℂ]
theorem hasDerivAt_exp (x : ℂ) : HasDerivAt exp (exp x) x := by
rw [hasDerivAt_iff_isLittleO_nhds_zero]
have : (1 : ℕ) < 2 := by norm_num
refine (IsBigO.of_bound ‖exp x‖ ?_).trans_isLittleO (isLittleO_pow_id this)
filter_upwards [Metric.ball_mem_nhds (0 : ℂ) zero_lt_one]
simp only [Metric.mem_ball, dist_zero_right, norm_pow]
exact fun z hz => exp_bound_sq x z hz.le
#align complex.has_deriv_at_exp Complex.hasDerivAt_exp
theorem differentiable_exp : Differentiable 𝕜 exp := fun x =>
(hasDerivAt_exp x).differentiableAt.restrictScalars 𝕜
#align complex.differentiable_exp Complex.differentiable_exp
theorem differentiableAt_exp {x : ℂ} : DifferentiableAt 𝕜 exp x :=
differentiable_exp x
#align complex.differentiable_at_exp Complex.differentiableAt_exp
@[simp]
theorem deriv_exp : deriv exp = exp :=
funext fun x => (hasDerivAt_exp x).deriv
#align complex.deriv_exp Complex.deriv_exp
@[simp]
theorem iter_deriv_exp : ∀ n : ℕ, deriv^[n] exp = exp
| 0 => rfl
| n + 1 => by rw [iterate_succ_apply, deriv_exp, iter_deriv_exp n]
#align complex.iter_deriv_exp Complex.iter_deriv_exp
| Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 64 | 73 | theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp := by |
-- Porting note: added `@` due to `∀ {n}` weirdness above
refine @(contDiff_all_iff_nat.2 fun n => ?_)
have : ContDiff ℂ (↑n) exp := by
induction' n with n ihn
· exact contDiff_zero.2 continuous_exp
· rw [contDiff_succ_iff_deriv]
use differentiable_exp
rwa [deriv_exp]
exact this.restrict_scalars 𝕜
| 2,135 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Finset Set
open scoped Topology
namespace Real
variable {x : ℝ}
| Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 34 | 39 | theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by |
have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
(hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
(ne_of_gt <| exp_pos _) <|
Eventually.mono (lt_mem_nhds hx) @exp_log
rwa [exp_log hx] at this
| 2,136 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Finset Set
open scoped Topology
namespace Real
variable {x : ℝ}
theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by
have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
(hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
(ne_of_gt <| exp_pos _) <|
Eventually.mono (lt_mem_nhds hx) @exp_log
rwa [exp_log hx] at this
#align real.has_strict_deriv_at_log_of_pos Real.hasStrictDerivAt_log_of_pos
| Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 42 | 47 | theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by |
cases' hx.lt_or_lt with hx hx
· convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
· ext y; exact (log_neg_eq_log y).symm
· field_simp [hx.ne]
· exact hasStrictDerivAt_log_of_pos hx
| 2,136 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Filter Finset Set
open scoped Topology
namespace Real
variable {x : ℝ}
theorem hasStrictDerivAt_log_of_pos (hx : 0 < x) : HasStrictDerivAt log x⁻¹ x := by
have : HasStrictDerivAt log (exp <| log x)⁻¹ x :=
(hasStrictDerivAt_exp <| log x).of_local_left_inverse (continuousAt_log hx.ne')
(ne_of_gt <| exp_pos _) <|
Eventually.mono (lt_mem_nhds hx) @exp_log
rwa [exp_log hx] at this
#align real.has_strict_deriv_at_log_of_pos Real.hasStrictDerivAt_log_of_pos
theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
cases' hx.lt_or_lt with hx hx
· convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
· ext y; exact (log_neg_eq_log y).symm
· field_simp [hx.ne]
· exact hasStrictDerivAt_log_of_pos hx
#align real.has_strict_deriv_at_log Real.hasStrictDerivAt_log
theorem hasDerivAt_log (hx : x ≠ 0) : HasDerivAt log x⁻¹ x :=
(hasStrictDerivAt_log hx).hasDerivAt
#align real.has_deriv_at_log Real.hasDerivAt_log
theorem differentiableAt_log (hx : x ≠ 0) : DifferentiableAt ℝ log x :=
(hasDerivAt_log hx).differentiableAt
#align real.differentiable_at_log Real.differentiableAt_log
theorem differentiableOn_log : DifferentiableOn ℝ log {0}ᶜ := fun _x hx =>
(differentiableAt_log hx).differentiableWithinAt
#align real.differentiable_on_log Real.differentiableOn_log
@[simp]
theorem differentiableAt_log_iff : DifferentiableAt ℝ log x ↔ x ≠ 0 :=
⟨fun h => continuousAt_log_iff.1 h.continuousAt, differentiableAt_log⟩
#align real.differentiable_at_log_iff Real.differentiableAt_log_iff
theorem deriv_log (x : ℝ) : deriv log x = x⁻¹ :=
if hx : x = 0 then by
rw [deriv_zero_of_not_differentiableAt (differentiableAt_log_iff.not_left.2 hx), hx, inv_zero]
else (hasDerivAt_log hx).deriv
#align real.deriv_log Real.deriv_log
@[simp]
theorem deriv_log' : deriv log = Inv.inv :=
funext deriv_log
#align real.deriv_log' Real.deriv_log'
| Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 78 | 81 | theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by |
suffices ContDiffOn ℝ ⊤ log {0}ᶜ from this.of_le le_top
refine (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 ?_
simp [differentiableOn_log, contDiffOn_inv]
| 2,136 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
| Mathlib/Data/Complex/ExponentialBounds.lean | 20 | 25 | theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by |
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
| 2,137 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_10 Real.exp_one_near_10
| Mathlib/Data/Complex/ExponentialBounds.lean | 28 | 33 | theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by |
apply exp_approx_start
iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
| 2,137 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_10 Real.exp_one_near_10
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
apply exp_approx_start
iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_20 Real.exp_one_near_20
theorem exp_one_gt_d9 : 2.7182818283 < exp 1 :=
lt_of_lt_of_le (by norm_num) (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
#align real.exp_one_gt_d9 Real.exp_one_gt_d9
theorem exp_one_lt_d9 : exp 1 < 2.7182818286 :=
lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) (by norm_num)
#align real.exp_one_lt_d9 Real.exp_one_lt_d9
| Mathlib/Data/Complex/ExponentialBounds.lean | 44 | 48 | theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by |
rw [exp_neg, lt_inv _ (exp_pos _)]
· refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num
· norm_num
| 2,137 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_10 Real.exp_one_near_10
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
apply exp_approx_start
iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_20 Real.exp_one_near_20
theorem exp_one_gt_d9 : 2.7182818283 < exp 1 :=
lt_of_lt_of_le (by norm_num) (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
#align real.exp_one_gt_d9 Real.exp_one_gt_d9
theorem exp_one_lt_d9 : exp 1 < 2.7182818286 :=
lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) (by norm_num)
#align real.exp_one_lt_d9 Real.exp_one_lt_d9
theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by
rw [exp_neg, lt_inv _ (exp_pos _)]
· refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num
· norm_num
#align real.exp_neg_one_gt_d9 Real.exp_neg_one_gt_d9
| Mathlib/Data/Complex/ExponentialBounds.lean | 51 | 55 | theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by |
rw [exp_neg, inv_lt (exp_pos _)]
· refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
norm_num
· norm_num
| 2,137 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 2244083 / 825552| ≤ 1 / 10 ^ 10 := by
apply exp_approx_start
iterate 13 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_10 Real.exp_one_near_10
theorem exp_one_near_20 : |exp 1 - 363916618873 / 133877442384| ≤ 1 / 10 ^ 20 := by
apply exp_approx_start
iterate 21 refine exp_1_approx_succ_eq (by norm_num1; rfl) (by norm_cast) ?_
norm_num1
refine exp_approx_end' _ (by norm_num1; rfl) _ (by norm_cast) (by simp) ?_
rw [_root_.abs_one, abs_of_pos] <;> norm_num1
#align real.exp_one_near_20 Real.exp_one_near_20
theorem exp_one_gt_d9 : 2.7182818283 < exp 1 :=
lt_of_lt_of_le (by norm_num) (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
#align real.exp_one_gt_d9 Real.exp_one_gt_d9
theorem exp_one_lt_d9 : exp 1 < 2.7182818286 :=
lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) (by norm_num)
#align real.exp_one_lt_d9 Real.exp_one_lt_d9
theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by
rw [exp_neg, lt_inv _ (exp_pos _)]
· refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num
· norm_num
#align real.exp_neg_one_gt_d9 Real.exp_neg_one_gt_d9
theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by
rw [exp_neg, inv_lt (exp_pos _)]
· refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
norm_num
· norm_num
#align real.exp_neg_one_lt_d9 Real.exp_neg_one_lt_d9
set_option tactic.skipAssignedInstances false in
| Mathlib/Data/Complex/ExponentialBounds.lean | 59 | 71 | theorem log_two_near_10 : |log 2 - 287209 / 414355| ≤ 1 / 10 ^ 10 := by |
suffices |log 2 - 287209 / 414355| ≤ 1 / 17179869184 + (1 / 10 ^ 10 - 1 / 2 ^ 34) by
norm_num1 at *
assumption
have t : |(2⁻¹ : ℝ)| = 2⁻¹ := by rw [abs_of_pos]; norm_num
have z := Real.abs_log_sub_add_sum_range_le (show |(2⁻¹ : ℝ)| < 1 by rw [t]; norm_num) 34
rw [t] at z
norm_num1 at z
rw [one_div (2 : ℝ), log_inv, ← sub_eq_add_neg, _root_.abs_sub_comm] at z
apply le_trans (_root_.abs_sub_le _ _ _) (add_le_add z _)
simp_rw [sum_range_succ]
norm_num
rw [abs_of_pos] <;> norm_num
| 2,137 |
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import ring_theory.polynomial.hermite.gaussian from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Polynomial
namespace Polynomial
| Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 40 | 55 | theorem deriv_gaussian_eq_hermite_mul_gaussian (n : ℕ) (x : ℝ) :
deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x =
(-1 : ℝ) ^ n * aeval x (hermite n) * Real.exp (-(x ^ 2 / 2)) := by |
rw [mul_assoc]
induction' n with n ih generalizing x
· rw [Function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul]
· replace ih : deriv^[n] _ = _ := _root_.funext ih
have deriv_gaussian :
deriv (fun y => Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2)) := by
-- porting note (#10745): was `simp [mul_comm, ← neg_mul]`
rw [deriv_exp (by simp)]; simp; ring
rw [Function.iterate_succ_apply', ih, deriv_const_mul_field, deriv_mul, pow_succ (-1 : ℝ),
deriv_gaussian, hermite_succ, map_sub, map_mul, aeval_X, Polynomial.deriv_aeval]
· ring
· apply Polynomial.differentiable_aeval
· apply DifferentiableAt.exp; simp -- Porting note: was just `simp`
| 2,138 |
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import ring_theory.polynomial.hermite.gaussian from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Polynomial
namespace Polynomial
theorem deriv_gaussian_eq_hermite_mul_gaussian (n : ℕ) (x : ℝ) :
deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x =
(-1 : ℝ) ^ n * aeval x (hermite n) * Real.exp (-(x ^ 2 / 2)) := by
rw [mul_assoc]
induction' n with n ih generalizing x
· rw [Function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul]
· replace ih : deriv^[n] _ = _ := _root_.funext ih
have deriv_gaussian :
deriv (fun y => Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2)) := by
-- porting note (#10745): was `simp [mul_comm, ← neg_mul]`
rw [deriv_exp (by simp)]; simp; ring
rw [Function.iterate_succ_apply', ih, deriv_const_mul_field, deriv_mul, pow_succ (-1 : ℝ),
deriv_gaussian, hermite_succ, map_sub, map_mul, aeval_X, Polynomial.deriv_aeval]
· ring
· apply Polynomial.differentiable_aeval
· apply DifferentiableAt.exp; simp -- Porting note: was just `simp`
#align polynomial.deriv_gaussian_eq_hermite_mul_gaussian Polynomial.deriv_gaussian_eq_hermite_mul_gaussian
| Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 58 | 64 | theorem hermite_eq_deriv_gaussian (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2)) := by |
rw [deriv_gaussian_eq_hermite_mul_gaussian]
field_simp [Real.exp_ne_zero]
rw [← @smul_eq_mul ℝ _ ((-1) ^ n), ← inv_smul_eq_iff₀, mul_assoc, smul_eq_mul, ← inv_pow, ←
neg_inv, inv_one]
exact pow_ne_zero _ (by norm_num)
| 2,138 |
import Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import ring_theory.polynomial.hermite.gaussian from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Polynomial
namespace Polynomial
theorem deriv_gaussian_eq_hermite_mul_gaussian (n : ℕ) (x : ℝ) :
deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x =
(-1 : ℝ) ^ n * aeval x (hermite n) * Real.exp (-(x ^ 2 / 2)) := by
rw [mul_assoc]
induction' n with n ih generalizing x
· rw [Function.iterate_zero_apply, pow_zero, one_mul, hermite_zero, C_1, map_one, one_mul]
· replace ih : deriv^[n] _ = _ := _root_.funext ih
have deriv_gaussian :
deriv (fun y => Real.exp (-(y ^ 2 / 2))) x = -x * Real.exp (-(x ^ 2 / 2)) := by
-- porting note (#10745): was `simp [mul_comm, ← neg_mul]`
rw [deriv_exp (by simp)]; simp; ring
rw [Function.iterate_succ_apply', ih, deriv_const_mul_field, deriv_mul, pow_succ (-1 : ℝ),
deriv_gaussian, hermite_succ, map_sub, map_mul, aeval_X, Polynomial.deriv_aeval]
· ring
· apply Polynomial.differentiable_aeval
· apply DifferentiableAt.exp; simp -- Porting note: was just `simp`
#align polynomial.deriv_gaussian_eq_hermite_mul_gaussian Polynomial.deriv_gaussian_eq_hermite_mul_gaussian
theorem hermite_eq_deriv_gaussian (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x / Real.exp (-(x ^ 2 / 2)) := by
rw [deriv_gaussian_eq_hermite_mul_gaussian]
field_simp [Real.exp_ne_zero]
rw [← @smul_eq_mul ℝ _ ((-1) ^ n), ← inv_smul_eq_iff₀, mul_assoc, smul_eq_mul, ← inv_pow, ←
neg_inv, inv_one]
exact pow_ne_zero _ (by norm_num)
#align polynomial.hermite_eq_deriv_gaussian Polynomial.hermite_eq_deriv_gaussian
| Mathlib/RingTheory/Polynomial/Hermite/Gaussian.lean | 67 | 70 | theorem hermite_eq_deriv_gaussian' (n : ℕ) (x : ℝ) : aeval x (hermite n) =
(-1 : ℝ) ^ n * deriv^[n] (fun y => Real.exp (-(y ^ 2 / 2))) x * Real.exp (x ^ 2 / 2) := by |
rw [hermite_eq_deriv_gaussian, Real.exp_neg]
field_simp [Real.exp_ne_zero]
| 2,138 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Filter Real
open scoped Classical Topology NNReal
noncomputable def gronwallBound (δ K ε x : ℝ) : ℝ :=
if K = 0 then δ + ε * x else δ * exp (K * x) + ε / K * (exp (K * x) - 1)
#align gronwall_bound gronwallBound
theorem gronwallBound_K0 (δ ε : ℝ) : gronwallBound δ 0 ε = fun x => δ + ε * x :=
funext fun _ => if_pos rfl
set_option linter.uppercaseLean3 false in
#align gronwall_bound_K0 gronwallBound_K0
theorem gronwallBound_of_K_ne_0 {δ K ε : ℝ} (hK : K ≠ 0) :
gronwallBound δ K ε = fun x => δ * exp (K * x) + ε / K * (exp (K * x) - 1) :=
funext fun _ => if_neg hK
set_option linter.uppercaseLean3 false in
#align gronwall_bound_of_K_ne_0 gronwallBound_of_K_ne_0
| Mathlib/Analysis/ODE/Gronwall.lean | 59 | 70 | theorem hasDerivAt_gronwallBound (δ K ε x : ℝ) :
HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x := by |
by_cases hK : K = 0
· subst K
simp only [gronwallBound_K0, zero_mul, zero_add]
convert ((hasDerivAt_id x).const_mul ε).const_add δ
rw [mul_one]
· simp only [gronwallBound_of_K_ne_0 hK]
convert (((hasDerivAt_id x).const_mul K).exp.const_mul δ).add
((((hasDerivAt_id x).const_mul K).exp.sub_const 1).const_mul (ε / K)) using 1
simp only [id, mul_add, (mul_assoc _ _ _).symm, mul_comm _ K, mul_div_cancel₀ _ hK]
ring
| 2,139 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Filter Real
open scoped Classical Topology NNReal
noncomputable def gronwallBound (δ K ε x : ℝ) : ℝ :=
if K = 0 then δ + ε * x else δ * exp (K * x) + ε / K * (exp (K * x) - 1)
#align gronwall_bound gronwallBound
theorem gronwallBound_K0 (δ ε : ℝ) : gronwallBound δ 0 ε = fun x => δ + ε * x :=
funext fun _ => if_pos rfl
set_option linter.uppercaseLean3 false in
#align gronwall_bound_K0 gronwallBound_K0
theorem gronwallBound_of_K_ne_0 {δ K ε : ℝ} (hK : K ≠ 0) :
gronwallBound δ K ε = fun x => δ * exp (K * x) + ε / K * (exp (K * x) - 1) :=
funext fun _ => if_neg hK
set_option linter.uppercaseLean3 false in
#align gronwall_bound_of_K_ne_0 gronwallBound_of_K_ne_0
theorem hasDerivAt_gronwallBound (δ K ε x : ℝ) :
HasDerivAt (gronwallBound δ K ε) (K * gronwallBound δ K ε x + ε) x := by
by_cases hK : K = 0
· subst K
simp only [gronwallBound_K0, zero_mul, zero_add]
convert ((hasDerivAt_id x).const_mul ε).const_add δ
rw [mul_one]
· simp only [gronwallBound_of_K_ne_0 hK]
convert (((hasDerivAt_id x).const_mul K).exp.const_mul δ).add
((((hasDerivAt_id x).const_mul K).exp.sub_const 1).const_mul (ε / K)) using 1
simp only [id, mul_add, (mul_assoc _ _ _).symm, mul_comm _ K, mul_div_cancel₀ _ hK]
ring
#align has_deriv_at_gronwall_bound hasDerivAt_gronwallBound
| Mathlib/Analysis/ODE/Gronwall.lean | 73 | 76 | theorem hasDerivAt_gronwallBound_shift (δ K ε x a : ℝ) :
HasDerivAt (fun y => gronwallBound δ K ε (y - a)) (K * gronwallBound δ K ε (x - a) + ε) x := by |
convert (hasDerivAt_gronwallBound δ K ε _).comp x ((hasDerivAt_id x).sub_const a) using 1
rw [id, mul_one]
| 2,139 |
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