File size: 2,693 Bytes
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 | import OAI.Probability.InvariantIsing.Fields.FieldAffineScalarBridge
import OAI.Probability.InvariantIsing.Fields.FieldScalarSquareOrder
namespace OAI
/-! Evenness, monotonicity and the unit spin bound for the actual affine
log-cosh recursion. These hypotheses of the mixed-order argument are
discharged from the explicit terminal function. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped NNReal
namespace InvariantIsing
theorem fieldAffineFamily_shape (I : Set ℝ) (hI : IsOpen I) (L : List FieldAffineStep)
(hL : ∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t)
(hζ : ∀ av ∈ L, 0 < av.exponent) (t : ℝ) :
Function.Even (fun z => (fieldAffineFamily I hI L hL).U (t, z)) ∧
Function.Odd (fun z => (fieldAffineFamily I hI L hL).X (t, z)) ∧
MonotoneOn (fun z => (fieldAffineFamily I hI L hL).X (t, z)) (Ici 0) ∧
(∀ z ∈ Ici (0 : ℝ), 0 ≤ (fieldAffineFamily I hI L hL).X (t, z)) ∧
(∀ z, |(fieldAffineFamily I hI L hL).X (t, z)| ≤ 1) := by
have hi : ∀ av ∈ fieldAffineIncrements L t, 0 < av.1 := by
intro av hav
obtain ⟨bv, hb, rfl⟩ := List.mem_map.mp hav
exact hζ bv hb
have hF : Measurable (fun z : ℝ => Real.log (Real.cosh z)) := by fun_prop
have hFg : HasLinearGrowth (fun z : ℝ => Real.log (Real.cosh z)) :=
(fieldLogCoshFamily I).growth 0
have hFe : Function.Even (fun z : ℝ => Real.log (Real.cosh z)) := by
intro z
simp only [Real.cosh_neg]
have ha : Measurable (fun z : ℝ => Real.tanh z) := by
simp only [Real.tanh_eq]
fun_prop
have hao : Function.Odd Real.tanh := Real.tanh_neg
have ham : MonotoneOn Real.tanh (Ici 0) := field_tanh_monotone.monotoneOn _
have hap : ∀ z ∈ Ici (0 : ℝ), 0 ≤ Real.tanh z := fun _ hz => field_tanh_nonneg hz
have hv := fieldScalarValue_even (fieldAffineIncrements L t) hi hF hFg hFe
have hm := fieldScalarMean_shape (fieldAffineIncrements L t) hi hF hFg hFe
ha hao ham hap field_abs_tanh_le_one
have hb := (fieldScalarMean_regular (fieldAffineIncrements L t) hi hF hFg
ha field_abs_tanh_le_one).2
have hU : (fun z => (fieldAffineFamily I hI L hL).U (t, z)) =
fieldScalarValue (fieldAffineIncrements L t) (fun z => Real.log (Real.cosh z)) := by
rw [fieldAffineFamily_value I hI L hL]
exact fieldAffineValue_scalar L t
have hX := fieldAffineFamily_mean I hI L hL t
have hXp (z : ℝ) := congrFun hX z
refine ⟨?_, ?_, ?_, ?_, ?_⟩
· rw [hU]
exact hv
· rw [hX]
exact hm.1
· rw [hX]
exact hm.2.1
· intro z hz
rw [hXp]
exact hm.2.2 z hz
· intro z
rw [hXp]
exact hb z
end InvariantIsing
end
end OAI
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