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b317718 | 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 | import OAI.Probability.InvariantIsing.Arrays.ReplicaProductPrimitives
namespace OAI
/-! Symbols determine bounded measurable paths up to a constant. A common
right root removes this constant, including jumps accumulating at zero. -/
noncomputable section
open MeasureTheory Set Filter
open scoped Topology
namespace InvariantIsing
theorem equal_pathSymbols_constant_difference (a b : ℝ → ℝ)
(ha : Measurable a) (hb : Measurable b) {A B : ℝ}
(haB : ∀ u, |a u| ≤ A) (hbB : ∀ u, |b u| ≤ B) (da db : ℝ)
(heq : ∀ s ∈ Ioo (0 : ℝ) 1, pathSymbol da a s = pathSymbol db b s)
{s t : ℝ} (hs : s ∈ Ioo (0 : ℝ) 1) (ht : t ∈ Ioo (0 : ℝ) 1) :
a s - b s = a t - b t := by
let h : ℝ → ℝ := fun u => a u - b u
let D := da - db
let H : ℝ → ℝ := fun u => ∫ v in u..1, h v
let G : ℝ → ℝ := fun u => (D - H u) / u
have hhi (v w : ℝ) : IntervalIntegrable h volume v w :=
intervalIntegrable_of_measurable_abs_le (ha.sub hb)
(fun u => (abs_sub _ _).trans (add_le_add (haB u) (hbB u))) v w
have hai (v w : ℝ) : IntervalIntegrable a volume v w :=
intervalIntegrable_of_measurable_abs_le ha haB v w
have hbi (v w : ℝ) : IntervalIntegrable b volume v w :=
intervalIntegrable_of_measurable_abs_le hb hbB v w
have hH : Continuous H := continuous_tail_integral hhi 1
have hrel (u : ℝ) (hu : u ∈ Ioo (0 : ℝ) 1) : D - u * h u - H u = 0 := by
have he := heq u hu
unfold pathSymbol at he
dsimp only [D, H, h]
rw [intervalIntegral.integral_sub (hai u 1) (hbi u 1)]
linarith
have hid (u : ℝ) (hu : u ∈ Ioo (0 : ℝ) 1) : h u = G u := by
apply (eq_div_iff hu.1.ne').mpr
have he := hrel u hu
linarith
have hGd (u : ℝ) (hu : u ∈ Ioo (0 : ℝ) 1) : HasDerivAt G 0 u := by
have hGc : ContinuousAt G u := (continuousAt_const.sub hH.continuousAt).div
continuousAt_id hu.1.ne'
have hhc : ContinuousAt h u := hGc.congr_of_eventuallyEq (by
filter_upwards [isOpen_Ioo.mem_nhds hu] with v hv
exact hid v hv)
have hHd : HasDerivAt H (-h u) u := intervalIntegral.integral_hasDerivAt_left
(hhi u 1) (ha.sub hb).stronglyMeasurable.stronglyMeasurableAtFilter hhc
have hd := ((hasDerivAt_const u D).sub hHd).div (hasDerivAt_id u) hu.1.ne'
change HasDerivAt G (((0 - -h u) * u - (D - H u) * 1) / u ^ 2) u at hd
have hz : ((0 - -h u) * u - (D - H u) * 1) / u ^ 2 = 0 := by
apply div_eq_zero_iff.mpr
left
have he := hrel u hu
nlinarith
rw [hz] at hd
exact hd
have hc := isOpen_Ioo.is_const_of_deriv_eq_zero isPreconnected_Ioo
(fun u hu => (hGd u hu).differentiableAt.differentiableWithinAt)
(fun u hu => (hGd u hu).deriv) hs ht
exact (hid s hs).trans (hc.trans (hid t ht).symm)
theorem equal_pathSymbols_equal_roots (a b : ℝ → ℝ)
(ha : Measurable a) (hb : Measurable b) {A B : ℝ}
(haB : ∀ u, |a u| ≤ A) (hbB : ∀ u, |b u| ≤ B) (da db r : ℝ)
(heq : ∀ s ∈ Ioo (0 : ℝ) 1, pathSymbol da a s = pathSymbol db b s)
(hra : Tendsto a (𝓝[>] 0) (𝓝 r)) (hrb : Tendsto b (𝓝[>] 0) (𝓝 r)) :
da = db ∧ ∀ s ∈ Ioo (0 : ℝ) 1, a s = b s := by
let c := a (1 / 2) - b (1 / 2)
have hc (s : ℝ) (hs : s ∈ Ioo (0 : ℝ) 1) : a s - b s = c :=
equal_pathSymbols_constant_difference a b ha hb haB hbB da db heq hs (by norm_num)
have hce : (fun s => a s - b s) =ᶠ[𝓝[>] (0 : ℝ)] fun _ => c := by
filter_upwards [self_mem_nhdsWithin,
(eventually_lt_nhds (show (0 : ℝ) < 1 by norm_num)).filter_mono nhdsWithin_le_nhds]
with s hs hlt
exact hc s ⟨hs, hlt⟩
have hlim : Tendsto (fun s => a s - b s) (𝓝[>] 0) (𝓝 (0 : ℝ)) := by
simpa only [sub_self] using hra.sub hrb
have hc0 : c = 0 := tendsto_nhds_unique (tendsto_const_nhds.congr' hce.symm) hlim
have hab (s : ℝ) (hs : s ∈ Ioo (0 : ℝ) 1) : a s = b s := sub_eq_zero.mp ((hc s hs).trans hc0)
refine ⟨?_, hab⟩
have hi : (∫ u in (1 / 2 : ℝ)..1, a u) = ∫ u in (1 / 2 : ℝ)..1, b u := by
apply intervalIntegral.integral_congr_Ioo_of_le (by norm_num)
intro u hu
exact hab u ⟨by linarith [hu.1], hu.2⟩
have he := heq (1 / 2) (by norm_num)
unfold pathSymbol at he
rw [hab (1 / 2) (by norm_num), hi] at he
linarith
end InvariantIsing
end
end OAI
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