| import OAI.Probability.ContinuumTransition.DensityBounds |
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| namespace OAI |
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| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
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| lemma physical_coefficient_lower {t β m : ℝ} (ht : 0 < t) (hβ : 0 ≤ β) |
| (hm : 17576*kernelHeight*m ≤ t) : |
| β*t/1024 ≤ (β*((t^2/8-2197*t*kernelHeight*m/2)/64)/t) := by |
| apply (le_div_iff₀ ht).mpr |
| have hh := mul_le_mul_of_nonneg_left hm ht.le |
| have hh' : t^2/16 ≤ t^2/8-2197*t*kernelHeight*m/2 := by nlinarith only [hh] |
| have hh'' := mul_le_mul_of_nonneg_left hh' hβ |
| nlinarith only [hh''] |
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| lemma bounded_density_of_quadratic {t β h a H P d : ℝ} |
| (ht : 1 ≤ t) (hβ : 1/4 ≤ β) (hβ' : β ≤ 2) (hh : h ≤ 10) |
| (ha : 1024*a ≤ t) (hd : 0 ≤ d) (hP : 0 ≤ P) |
| (hH : β*t/1024*d^2-(β*t/2)*d-1/t ≤ H) |
| (he : P = h*d+β*a*d^2/2-H) : d ≤ 1048576 := by |
| have ht0 : 0 < t := by linarith |
| have hc : t/8192 ≤ β*t/1024-β*a/2 := by |
| have hh1 := mul_le_mul_of_nonneg_left ha (show 0 ≤ β by linarith) |
| have hh2 := mul_le_mul_of_nonneg_right hβ ht0.le |
| nlinarith only [hh1,hh2] |
| have hlin : h+β*t/2 ≤ 11*t := by |
| have hh1 := mul_le_mul_of_nonneg_right hβ' ht0.le |
| nlinarith only [hh,hh1,ht] |
| have hinv : 1/t ≤ t := by |
| apply (div_le_iff₀ ht0).mpr |
| nlinarith only [sq_nonneg (t-1),ht] |
| have hh0 := mul_le_mul_of_nonneg_right hc (sq_nonneg d) |
| have hh1 := mul_le_mul_of_nonneg_right hlin hd |
| have hquad : t*(d^2/8192-11*d-1) ≤ 0 := by nlinarith only [hh0,hh1,hinv,hH,hP,he] |
| have hquad' : d^2/8192-11*d-1 ≤ 0 := by |
| by_contra hn |
| have hh := mul_pos ht0 (lt_of_not_ge hn) |
| linarith only [hh,hquad] |
| by_contra hn |
| have hg : 1048576 < d := lt_of_not_ge hn |
| have hh := mul_nonneg hd (le_of_lt (sub_pos.mpr hg)) |
| nlinarith only [hquad',hg,hh] |
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| lemma meanField_ge_base {Q P : Parameter → ℝ} {a : ℝ} |
| (hP : IsMeanField Q a P) {x : Parameter} (hx : x ∈ pressureDomain) : Q x ≤ P x := by |
| have hh := (hP x hx).1 (mem_range_self (0:ℝ)) |
| simpa only [quadraticCost,shiftedField,mul_zero,add_zero,zero_pow (by decide : 2 ≠ 0),zero_div,sub_zero,Prod.mk.eta] using hh |
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| lemma active_pressure_gap {Q P : Parameter → ℝ} {H : ℝ → ℝ → ℝ} {a D : ℝ} |
| (hH : IsPartialConjugate Q H) |
| (hm : ∀ β, 0 < β → Monotone (fun h => Q (β,h))) |
| (hQ : ∀ β, 0 < β → ∀ h, IsLUB ((fun d => h*d-H β d) '' Ici 0) (Q (β,h))) |
| {x w : Parameter} (hx : x ∈ pressureDomain) (ha : 0 ≤ a) |
| (hw : w ∈ activeLifts Q a P x) (hd : w.2 ≤ D) : P x ≤ Q x+x.1*a*D^2/2 := by |
| have hd0 := active_density_nonneg hm hx hw |
| have he := active_canonical_eq hH hm hx hw |
| have hl := (hQ x.1 hx x.2).1 (mem_image_of_mem (fun d => x.2*d-H x.1 d) hd0) |
| have hs : w.2^2 ≤ D^2 := sq_le_sq₀ hd0 (hd0.trans hd) |>.mpr hd |
| have hmul := mul_le_mul_of_nonneg_left hs (mul_nonneg (le_of_lt hx) ha) |
| nlinarith only [he,hl,hmul] |
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| end ContinuumTemperature |
| end |
| end |
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| end OAI |
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