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f61e547 | 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 | import OAI.Probability.FreeIsing.Cavity.Pair
import OAI.Probability.FreeIsing.Estimates.MessageFunction
namespace OAI
noncomputable section
namespace Problem367
def cavitySiteMap (theta : ℝ) (p : ℝ × ℝ) : ℝ × ℝ :=
(p.1 + cavityMessage theta p.2, p.2 + cavityMessage theta p.1)
def cavityHomeomorph {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1) :
(ℝ × ℝ) ≃ₜ (ℝ × ℝ) where
toFun := cavitySiteMap theta
invFun := CavityPair.solve (lipschitzWith_cavityMessage h0 h1.le) h1
left_inv p := CavityPair.solve_eq _ _ ⟨rfl, rfl⟩
right_inv z := Prod.ext (CavityPair.solve_spec _ _ z).1 (CavityPair.solve_spec _ _ z).2
continuous_toFun := by
have hψ := (lipschitzWith_cavityMessage h0 h1.le).continuous
exact (continuous_fst.add (hψ.comp continuous_snd)).prodMk
(continuous_snd.add (hψ.comp continuous_fst))
continuous_invFun := CavityPair.continuous_solve _ _
def cavityFromSites {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
(z : ℝ × ℝ) : ℝ × ℝ := (cavityHomeomorph h0 h1).symm z
theorem cavityFromSites_spec {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
(z : ℝ × ℝ) :
(cavityFromSites h0 h1 z).1 + cavityMessage theta (cavityFromSites h0 h1 z).2 = z.1 ∧
(cavityFromSites h0 h1 z).2 + cavityMessage theta (cavityFromSites h0 h1 z).1 = z.2 :=
CavityPair.solve_spec (lipschitzWith_cavityMessage h0 h1.le) h1 z
theorem cavityFromSites_eq {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
{z p : ℝ × ℝ}
(hp : p.1 + cavityMessage theta p.2 = z.1 ∧
p.2 + cavityMessage theta p.1 = z.2) : cavityFromSites h0 h1 z = p :=
CavityPair.solve_eq (lipschitzWith_cavityMessage h0 h1.le) h1 hp
theorem continuous_cavityFromSites {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1) :
Continuous (cavityFromSites h0 h1) := (cavityHomeomorph h0 h1).symm.continuous
theorem measurable_cavityFromSites {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1) :
Measurable (cavityFromSites h0 h1) := (continuous_cavityFromSites h0 h1).measurable
theorem cavityFromSites_swap {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
(z : ℝ × ℝ) : cavityFromSites h0 h1 z.swap = (cavityFromSites h0 h1 z).swap :=
CavityPair.solve_swap (lipschitzWith_cavityMessage h0 h1.le) h1 z
theorem cavityFromSites_bound {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
(z : ℝ × ℝ) :
|(cavityFromSites h0 h1 z).1 - z.1| ≤ Real.artanh theta ∧
|(cavityFromSites h0 h1 z).2 - z.2| ≤ Real.artanh theta := by
have h := cavityFromSites_spec h0 h1 z
constructor
· rw [show (cavityFromSites h0 h1 z).1 - z.1 =
-cavityMessage theta (cavityFromSites h0 h1 z).2 by linarith [h.1], abs_neg]
exact cavityMessage_abs_le_artanh h0 h1 _
· rw [show (cavityFromSites h0 h1 z).2 - z.2 =
-cavityMessage theta (cavityFromSites h0 h1 z).1 by linarith [h.2], abs_neg]
exact cavityMessage_abs_le_artanh h0 h1 _
theorem cavityFromSites_neg {theta : ℝ} (h0 : 0 ≤ theta) (h1 : theta < 1)
(z : ℝ × ℝ) : cavityFromSites h0 h1 (-z) = -cavityFromSites h0 h1 z := by
apply cavityFromSites_eq h0 h1
have hs : theta ^ 2 ≤ 1 := by nlinarith
have h := cavityFromSites_spec h0 h1 z
change -((cavityFromSites h0 h1 z).1) +
cavityMessage theta (-((cavityFromSites h0 h1 z).2)) = -z.1 ∧
-((cavityFromSites h0 h1 z).2) +
cavityMessage theta (-((cavityFromSites h0 h1 z).1)) = -z.2
rw [cavityMessage_neg hs, cavityMessage_neg hs]
constructor <;> linarith [h.1, h.2]
end Problem367
end
end OAI
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