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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Fields.FieldSpatialParameterOrder
import OAI.Probability.InvariantIsing.Fields.FieldTangentOrder
import OAI.Probability.InvariantIsing.Fields.FieldAffineShape
namespace OAI
/-! The actual internal-height comparison in `fld:mixed`. The fixed
tail is the finite log-cosh recursion. Increasing one height adds its
parameter to one variance and subtracts it from the next variance. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped NNReal
namespace InvariantIsing
def fieldAdjacentTail (a b : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) :
FieldSmoothFamily (Ioo (-a) b) :=
fieldAffineFamily (Ioo (-a) b) isOpen_Ioo L (fun av hav _ _ => by
rw [hs av hav, zero_mul, add_zero]
exact hb av hav)
def fieldAdjacentInner (a b η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) :
FieldSmoothFamily (Ioo (-a) b) :=
(fieldAdjacentTail a b L hb hs).transform isOpen_Ioo b (-1) η (fun t ht => by
have h := ht.2
linarith)
def fieldAdjacentFamily (a b ζ η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) :
FieldSmoothFamily (Ioo (-a) b) :=
(fieldAdjacentInner a b η L hb hs).transform isOpen_Ioo a 1 ζ (fun t ht => by
have h := ht.1
linarith)
lemma fieldAdjacentTail_stationary (a b : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0) (p : ℝ × ℝ) :
(fieldAdjacentTail a b L hb hs).T p = 0 :=
fieldAffineFamily_stationary (Ioo (-a) b) isOpen_Ioo L _ hs p
lemma fieldAdjacentInner_shape (a b η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0)
(hη : 0 < η) (hζ : ∀ av ∈ L, 0 < av.exponent) (t : ℝ) :
let G := fieldAdjacentInner a b η L hb hs
Function.Even (fun z => G.U (t, z)) ∧
Function.Odd (fun z => G.X (t, z)) ∧
MonotoneOn (fun z => G.X (t, z)) (Ici 0) ∧
(∀ z ∈ Ici (0 : ℝ), 0 ≤ G.X (t, z)) ∧ (∀ z, |G.X (t, z)| ≤ 1) := by
let av : FieldAffineStep := ⟨b, -1, η⟩
have hv : ∀ v ∈ av :: L, ∀ q ∈ Ioo (-a) b, 0 < v.base + v.slope * q := by
intro v hv q hq
rcases List.mem_cons.mp hv with rfl | hv
· dsimp only [av]
linarith [hq.2]
· rw [hs v hv, zero_mul, add_zero]
exact hb v hv
have he : ∀ v ∈ av :: L, 0 < v.exponent := by
intro v hv
rcases List.mem_cons.mp hv with rfl | hv
· exact hη
· exact hζ v hv
change Function.Even (fun z => (fieldAffineFamily (Ioo (-a) b) isOpen_Ioo (av :: L) hv).U (t, z)) ∧ _
exact fieldAffineFamily_shape (Ioo (-a) b) isOpen_Ioo (av :: L) hv he t
lemma fieldAdjacentFamily_shape (a b ζ η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0)
(hζ : 0 < ζ) (hη : 0 < η) (htail : ∀ av ∈ L, 0 < av.exponent) (t : ℝ) :
let G := fieldAdjacentFamily a b ζ η L hb hs
Function.Even (fun z => G.U (t, z)) ∧
Function.Odd (fun z => G.X (t, z)) ∧
MonotoneOn (fun z => G.X (t, z)) (Ici 0) ∧
(∀ z ∈ Ici (0 : ℝ), 0 ≤ G.X (t, z)) ∧ (∀ z, |G.X (t, z)| ≤ 1) := by
let av : FieldAffineStep := ⟨a, 1, ζ⟩
let bv : FieldAffineStep := ⟨b, -1, η⟩
have hv : ∀ v ∈ av :: bv :: L, ∀ q ∈ Ioo (-a) b, 0 < v.base + v.slope * q := by
intro v hv q hq
rcases List.mem_cons.mp hv with rfl | hv
· dsimp only [av]
linarith [hq.1]
· rcases List.mem_cons.mp hv with rfl | hv
· dsimp only [bv]
linarith [hq.2]
· rw [hs v hv, zero_mul, add_zero]
exact hb v hv
have he : ∀ v ∈ av :: bv :: L, 0 < v.exponent := by
intro v hv
rcases List.mem_cons.mp hv with rfl | hv
· exact hζ
· rcases List.mem_cons.mp hv with rfl | hv
· exact hη
· exact htail v hv
change Function.Even (fun z => (fieldAffineFamily (Ioo (-a) b) isOpen_Ioo (av :: bv :: L) hv).U (t, z)) ∧ _
exact fieldAffineFamily_shape (Ioo (-a) b) isOpen_Ioo (av :: bv :: L) hv he t
theorem fieldAdjacentFamily_tangent_antitone (a b ζ η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0)
(hη : 0 < η) (hζη : ζ ≤ η) (htail : ∀ av ∈ L, 0 < av.exponent)
{t : ℝ} (ht : t ∈ Ioo (-a) b) :
AntitoneOn (fun z => (fieldAdjacentFamily a b ζ η L hb hs).T (t, z)) (Ici 0) := by
let F := fieldAdjacentTail a b L hb hs
let G := fieldAdjacentInner a b η L hb hs
have hg := fieldAdjacentInner_shape a b η L hb hs hη htail t
have hU : Measurable (fun z => G.U (t, z)) := G.mU.comp (by fun_prop)
have hX : Measurable (fun z => G.X (t, z)) := G.mX.comp (by fun_prop)
have hshape := field_negative_square_average_shape (a + t) ζ ((η - ζ) / 2)
(by linarith) hU hg.1 (G.growth t) hX hg.2.1 hg.2.2.1 hg.2.2.2.1 hg.2.2.2.2
have heq : (fun z => (fieldAdjacentFamily a b ζ η L hb hs).T (t, z)) =
(fun z => -((η - ζ) / 2) * gaussianTiltAverage (a + t) ζ
(fun y => G.U (t, y)) (fun y => (G.X (t, y)) ^ 2) z) := by
funext z
have h := F.adjacent_tangent isOpen_Ioo a b ζ η
(fun q hq => by linarith [hq.1]) (fun q hq => by linarith [hq.2])
(p := (t, z)) ht (fun y => fieldAdjacentTail_stationary a b L hb hs (t, y))
simpa only [F, G, fieldAdjacentFamily, fieldAdjacentInner, neg_div] using h
rw [heq]
exact hshape.2
theorem fieldAdjacent_earlier_squares_antitone (a b ζ η : ℝ) (L : List FieldAffineStep)
(hb : ∀ av ∈ L, 0 < av.base) (hs : ∀ av ∈ L, av.slope = 0)
(hζ : 0 < ζ) (hζη : ζ ≤ η) (htail : ∀ av ∈ L, 0 < av.exponent)
(P : List (ℝ × ℝ≥0)) (hP : ∀ av ∈ P, 0 < av.1)
{t s : ℝ} (ht : t ∈ Ioo (-a) b) (hss : s ∈ Ioo (-a) b) (hts : t ≤ s) :
let G := fieldAdjacentFamily a b ζ η L hb hs
∀ i z, fieldScalarSquares P (fun y => G.U (s, y)) (fun y => G.X (s, y)) i z ≤
fieldScalarSquares P (fun y => G.U (t, y)) (fun y => G.X (t, y)) i z := by
have hη : 0 < η := lt_of_lt_of_le hζ hζη
let G := fieldAdjacentFamily a b ζ η L hb hs
have hshape (q : ℝ) := fieldAdjacentFamily_shape a b ζ η L hb hs hζ hη htail q
exact G.earlier_squares_antitone_parameter (convex_Ioo (-a) b)
(fun q hq => fieldAdjacentFamily_tangent_antitone a b ζ η L hb hs hη hζη htail hq)
(fun q _ => (hshape q).1) (fun q _ => (hshape q).2.1)
(fun q _ => (hshape q).2.2.1) (fun q _ => (hshape q).2.2.2.1)
(fun p => (hshape p.1).2.2.2.2 p.2) P hP ht hss hts
end InvariantIsing
end
end OAI
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