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a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Pressure.FiniteTerminal
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology BigOperators BoundedContinuousFunction ContDiff
namespace SphericalPerceptronFreeEnergy
def Jet3.addConstant (g : Jet3) (c : ℝ) : Jet3 where
f:=g.f+BoundedContinuousFunction.const ℝ c
d1:=g.d1
d2:=g.d2
d3:=g.d3
has1 x:=by
change HasDerivAt (fun y => g.f y + c) (g.d1 x) x
exact (g.has1 x).add_const c
has2:=g.has2
has3:=g.has3
def Jet3.negative (g : Jet3) : Jet3 where
f:=-g.f
d1:=-g.d1
d2:=-g.d2
d3:=-g.d3
has1 x:=by simpa using (g.has1 x).neg
has2 x:=by simpa using (g.has2 x).neg
has3 x:=by simpa using (g.has3 x).neg
lemma exists_smooth_lower_envelope (f : ℝ→ᵇℝ) {R ε : ℝ} (hR : 0<R) (hε : 0<ε) :
∃ g : Jet3,
HasCompactSupport (g.d1 : ℝ→ℝ) ∧ HasCompactSupport (g.d2 : ℝ→ℝ) ∧
HasCompactSupport (g.d3 : ℝ→ℝ) ∧ ‖g.f‖≤‖f‖+2*ε ∧
(∀ x : ℝ,g.f x≤f x) ∧ (∀ x : ℝ,|x|≤R→|g.f x-f x|≤2*ε) := by
let K:=‖f‖
let ψ : ContDiffBump (0:ℝ):=⟨R,R+1,hR,by linarith⟩
let F : ℝ→ℝ:=fun x=>ψ x*(f x+K)
have hF : Continuous F:=ψ.continuous.mul (f.continuous.add continuous_const)
have hFs : HasCompactSupport F:=ψ.hasCompactSupport.mul_right
obtain ⟨g,hg,hgf,hgs⟩:=hF.exists_contDiff_approx (⊤:ℕ∞)
(ε:=fun _=>ε) continuous_const (fun _=>hε)
have hg' : ContDiff ℝ ∞ g:=hg
have hs : HasCompactSupport g:=hFs.mono hgs
let J:=(Jet3.ofCompactSmooth g hg' hs).addConstant (-K-ε)
have hJ (x : ℝ) : J.f x=g x-K-ε := by simp [J,Jet3.addConstant]; ring
have hzero (x : ℝ) : 0≤F x := by
have hf:= (abs_le.mp (f.norm_coe_le_norm x)).1
dsimp [F,K]
exact mul_nonneg ψ.nonneg (by linarith)
have hle (x : ℝ) : F x≤f x+K :=
mul_le_of_le_one_left (by have hf:=(abs_le.mp (f.norm_coe_le_norm x)).1; dsimp [K]; linarith) ψ.le_one
have herr (x : ℝ) : |g x-F x|<ε := by simpa only [Real.dist_eq] using hgf x
have hlower (x : ℝ) : J.f x≤f x := by
rw [hJ]
have hh:=(abs_lt.mp (herr x)).2
linarith [hle x]
refine ⟨J,?_,?_,?_,?_,hlower,?_⟩
· change HasCompactSupport (deriv g)
exact hs.deriv
· change HasCompactSupport (deriv (deriv g))
exact hs.deriv.deriv
· change HasCompactSupport (deriv (deriv (deriv g)))
exact hs.deriv.deriv.deriv
· apply (BoundedContinuousFunction.norm_le (by positivity : 0≤‖f‖+2*ε)).mpr
intro x
rw [Real.norm_eq_abs]
apply abs_le.mpr
constructor
· rw [hJ]
have hh:=(abs_lt.mp (herr x)).1
dsimp [K] at *
linarith [hzero x]
· have hh:=(abs_le.mp (f.norm_coe_le_norm x)).2
linarith [hlower x]
· intro x hx
have hψ : ψ x=1:=ψ.one_of_mem_closedBall (by simpa only [Metric.mem_closedBall,Real.dist_eq,sub_zero] using hx)
have he : F x=f x+K:=by simp [F,hψ]
rw [hJ]
have hh:=herr x
rw [he] at hh
exact abs_le.mpr ⟨by linarith [(abs_lt.mp hh).1],by linarith [(abs_lt.mp hh).2]⟩
lemma exists_smooth_lower_sequence (f : ℝ→ᵇℝ) :
∃ g : ℕ→Jet3,
(∀ n,HasCompactSupport ((g n).d1 : ℝ→ℝ) ∧ HasCompactSupport ((g n).d2 : ℝ→ℝ) ∧
HasCompactSupport ((g n).d3 : ℝ→ℝ)) ∧
(∀ n,‖(g n).f‖≤‖f‖+1) ∧ (∀ n x,(g n).f x≤f x) ∧
∀ R : ℝ,0<R→∀ ε : ℝ,0<ε→∃ N : ℕ,∀ n≥N,∀ x : ℝ,|x|≤R→|(g n).f x-f x|≤ε := by
have hx (n : ℕ):=exists_smooth_lower_envelope f
(by positivity : 0<(n:ℝ)+1) (by positivity : 0<(1:ℝ)/(2*((n:ℝ)+1)))
choose g h1 h2 h3 hn hlo he using hx
refine ⟨g,(fun n=>⟨h1 n,h2 n,h3 n⟩),?_,hlo,?_⟩
· intro n
have hh : 2*((1:ℝ)/(2*((n:ℝ)+1)))=1/((n:ℝ)+1) := by field_simp
have hle : (1:ℝ)/((n:ℝ)+1)≤1 := (div_le_one (by positivity)).mpr (by linarith [Nat.cast_nonneg (α:=ℝ) n])
have hi:=hn n
rw [hh] at hi
exact hi.trans (by linarith)
· intro R hR ε hε
obtain ⟨N1,hN1⟩:=exists_nat_ge R
obtain ⟨N2,hN2⟩:=exists_nat_one_div_lt hε
refine ⟨max N1 N2,fun n hn x hx=>?_⟩
have hN1' : N1≤n:=(le_max_left _ _).trans hn
have hN2' : N2≤n:=(le_max_right _ _).trans hn
have hx' : |x|≤(n:ℝ)+1 := by
have hc : (N1:ℝ)≤n:=by exact_mod_cast hN1'
linarith
have hh:=he n x hx'
have heq : 2*((1:ℝ)/(2*((n:ℝ)+1)))=1/((n:ℝ)+1) := by field_simp
rw [heq] at hh
exact hh.trans ((one_div_le_one_div_of_le (by positivity) (by exact_mod_cast Nat.add_le_add_right hN2' 1)).trans hN2.le)
lemma exists_smooth_upper_sequence (f : ℝ→ᵇℝ) :
∃ g : ℕ→Jet3,
(∀ n,HasCompactSupport ((g n).d1 : ℝ→ℝ) ∧ HasCompactSupport ((g n).d2 : ℝ→ℝ) ∧
HasCompactSupport ((g n).d3 : ℝ→ℝ)) ∧
(∀ n,‖(g n).f‖≤‖f‖+1) ∧ (∀ n x,f x≤(g n).f x) ∧
∀ R : ℝ,0<R→∀ ε : ℝ,0<ε→∃ N : ℕ,∀ n≥N,∀ x : ℝ,|x|≤R→|(g n).f x-f x|≤ε := by
obtain ⟨g,hs,hn,hlo,he⟩:=exists_smooth_lower_sequence (-f)
refine ⟨fun n=>(g n).negative,fun n=>⟨(hs n).1.neg,(hs n).2.1.neg,(hs n).2.2.neg⟩,?_,?_,?_⟩
· intro n
simpa only [Jet3.negative,norm_neg] using hn n
· intro n x
have h:=hlo n x
change f x≤-(g n).f x
change (g n).f x≤-f x at h
linarith
· intro R hR ε hε
obtain ⟨N,hN⟩:=he R hR ε hε
refine ⟨N,fun n hn x hx=>?_⟩
have h:=hN n hn x hx
change |-(g n).f x-f x|≤ε
change |(g n).f x- -f x|≤ε at h
rw [show -(g n).f x-f x = -((g n).f x+f x) by ring,abs_neg]
simpa only [sub_neg_eq_add] using h
end SphericalPerceptronFreeEnergy
end
end OAI
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