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59d1c80 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 | import Mathlib
import OAI.Geometry.IntegralFillings.Model
namespace OAI
section
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
namespace SharpIntegralFillings
attribute [local instance] Classical.propDecidable
universe u
section Foundations
variable {X : Type u} [MetricSpace X] [MeasurableSpace X]
namespace BoundedLip
omit [MeasurableSpace X] in
lemma continuous {b : X → ℝ} (hb : BoundedLip b) : Continuous b :=
hb.1.choose_spec.continuous
omit [MeasurableSpace X] in
lemma const (a : ℝ) : BoundedLip (fun _ : X => a) := by
exact ⟨⟨0, LipschitzWith.const a⟩, |a|, fun _ => le_rfl⟩
omit [MeasurableSpace X] in
lemma add {b c : X → ℝ} (hb : BoundedLip b) (hc : BoundedLip c) :
BoundedLip (fun x => b x + c x) := by
obtain ⟨⟨Kb, hKb⟩, Mb, hMb⟩ := hb
obtain ⟨⟨Kc, hKc⟩, Mc, hMc⟩ := hc
refine ⟨⟨Kb + Kc, hKb.add hKc⟩, Mb + Mc, ?_⟩
intro x
exact (abs_add_le _ _).trans (add_le_add (hMb x) (hMc x))
omit [MeasurableSpace X] in
lemma const_mul (a : ℝ) {b : X → ℝ} (hb : BoundedLip b) :
BoundedLip (fun x => a * b x) := by
obtain ⟨⟨K, hK⟩, M, hM⟩ := hb
refine ⟨⟨⟨|a|, abs_nonneg a⟩ * K, ?_⟩, |a| * M, ?_⟩
· apply lipschitzWith_iff_dist_le_mul.mpr
intro x y
change dist (a * b x) (a * b y) ≤ (|a| * (K : ℝ)) * dist x y
rw [Real.dist_eq, ← mul_sub, abs_mul]
calc |a| * |b x - b y| ≤ |a| * ((K : ℝ) * dist x y) :=
mul_le_mul_of_nonneg_left (hK.dist_le_mul x y) (abs_nonneg a)
_ = _ := by ring
· intro x
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (hM x) (abs_nonneg a)
end BoundedLip
lemma isMetricCurrent_zero (k : ℕ) : IsMetricCurrent (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) where
offDomain := by intros; rfl
linearFirst := by intros; simp
linearCoord := by intros; simp
sequentialContinuity := by intros; exact tendsto_const_nhds
locality := by intros; rfl
finiteMass := by
refine ⟨0, inferInstance, ?_⟩
intro b π hb hπ
simp
lemma mass_nonneg {k : ℕ} (T : Functional X k) : 0 ≤ mass T := by
apply Real.sInf_nonneg
rintro r ⟨μ, _, _, rfl⟩
exact ENNReal.toReal_nonneg
lemma mass_le_measure {k : ℕ} {T : Functional X k} {μ : Measure X}
(hμ : IsFiniteMeasure μ) (hT : Controls T μ) : mass T ≤ μ.real univ := by
apply csInf_le
· refine ⟨0, ?_⟩
rintro r ⟨ν, _, _, rfl⟩
exact ENNReal.toReal_nonneg
· exact ⟨μ, hμ, hT, rfl⟩
@[simp] lemma mass_zero (k : ℕ) : mass (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) = 0 := by
apply le_antisymm
· have h : Controls (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) (0 : Measure X) := by
intro b π hb hπ
simp
simpa using mass_le_measure (μ := (0 : Measure X)) inferInstance h
· exact mass_nonneg _
omit [MeasurableSpace X] in
@[simp] lemma boundarySucc_zero (k : ℕ) :
boundarySucc (fun (_ : X → ℝ) (_ : Fin (k+1) → X → ℝ) => 0) = fun _ _ => 0 := by
funext b π
simp [boundarySucc]
omit [MeasurableSpace X] in
@[simp] lemma boundary_zero (k : ℕ) :
boundary (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) = fun _ _ => 0 := by
cases k with
| zero => rfl
| succ k => exact boundarySucc_zero k
noncomputable def emptyChart (k : ℕ) : IntegerChart X k where
domain := ∅
borel := MeasurableSet.empty
bounded := Bornology.isBounded_empty
param := fun z => False.elim z.property
bilipschitz := by
refine ⟨0, 0, ?_, ?_⟩
· apply lipschitzWith_iff_dist_le_mul.mpr
intro a b
exact False.elim a.property
· intro a b
exact False.elim a.property
multiplicity := fun _ => 0
integrable := by simp
omit [MeasurableSpace X] in
@[simp] lemma emptyChart_image (k : ℕ) : (emptyChart (X := X) k).image = ∅ := by
ext x
simp [IntegerChart.image, emptyChart]
omit [MeasurableSpace X] in
@[simp] lemma emptyChart_action (k : ℕ) :
(emptyChart (X := X) k).action = fun _ _ => 0 := by
funext b π
simp [IntegerChart.action, emptyChart]
lemma integerRectifiable_zero (k : ℕ) :
IntegerRectifiable (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) := by
refine ⟨fun _ => emptyChart k, ?_, ?_, ?_, ?_⟩
· intro i j hij
simp
· intro i
rw [emptyChart_action]
exact isMetricCurrent_zero k
· simp only [emptyChart_action, mass_zero]
exact summable_zero
· intro b π
simp
lemma isIntegral_zero (k : ℕ) :
IsIntegral k (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) := by
cases k with
| zero => exact ⟨isMetricCurrent_zero 0, integerRectifiable_zero 0⟩
| succ k =>
refine ⟨isMetricCurrent_zero _, integerRectifiable_zero _, ?_, ?_⟩
· rw [boundarySucc_zero]
exact isMetricCurrent_zero k
· rw [boundarySucc_zero]
exact integerRectifiable_zero k
omit [MeasurableSpace X] in
lemma compactlySupported_zero (k : ℕ) :
CompactlySupported (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) :=
⟨∅, isCompact_empty, fun _ _ _ _ => rfl⟩
omit [MeasurableSpace X] in
lemma isCycle_zero (k : ℕ) : IsCycle (fun (_ : X → ℝ) (_ : Fin k → X → ℝ) => 0) :=
boundary_zero k
end Foundations
end SharpIntegralFillings
end
end OAI
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