| import OAI.Probability.ContinuumTransition.PhysicalUniform |
|
|
| namespace OAI |
|
|
| section |
| open scoped Topology |
| open Set Filter Metric |
| noncomputable section |
| namespace ContinuumTemperature |
|
|
| def threeCore (p : ℝ) (x : Parameter) : ℝ := max 0 (max (p*x.2) (4*p*x.2-9*p)) |
| def threeMeanField (p a : ℝ) (x : Parameter) : ℝ := |
| max 0 (max (p*x.2+x.1*a*p^2/2) (4*p*x.2-9*p+x.1*a*(4*p)^2/2)) |
| def firstMass (p : ℝ) : ℝ := 3/(2*p) |
| def initialPoint : Parameter := (1,-3/4) |
|
|
| lemma threeCore_convex (p : ℝ) : ConvexOn ℝ pressureDomain (threeCore p) := by |
| have hdom : Convex ℝ pressureDomain := by |
| change Convex ℝ {x : Parameter | 0 < x.1} |
| exact (convex_Ioi (0 : ℝ)).linear_preimage (LinearMap.fst ℝ ℝ ℝ) |
| apply (convexOn_const (0 : ℝ) hdom).sup |
| apply ConvexOn.sup |
| · refine ⟨hdom,?_⟩ |
| intro x hx y hy a b ha hb hab |
| dsimp |
| nlinarith |
| · refine ⟨hdom,?_⟩ |
| intro x hx y hy a b ha hb hab |
| dsimp |
| have hh := congrArg (fun z : ℝ => z*(9*p)) hab |
| nlinarith only [hh] |
|
|
| lemma three_meanField {p a : ℝ} (ha : 0 ≤ a) : IsMeanField (threeCore p) a (threeMeanField p a) := by |
| intro x hx |
| have hcoef : 0 ≤ x.1*a := mul_nonneg (le_of_lt hx) ha |
| have h0 : ∀ d : ℝ, -(x.1*a*d^2/2) ≤ 0 := fun d => neg_nonpos.mpr (div_nonneg (mul_nonneg hcoef (sq_nonneg d)) (by norm_num)) |
| have h1 : ∀ d : ℝ, p*(x.2+x.1*a*d)-x.1*a*d^2/2 ≤ p*x.2+x.1*a*p^2/2 := by |
| intro d |
| have hh := mul_nonneg hcoef (sq_nonneg (d-p)) |
| nlinarith only [hh] |
| have h2 : ∀ d : ℝ, 4*p*(x.2+x.1*a*d)-9*p-x.1*a*d^2/2 ≤ 4*p*x.2-9*p+x.1*a*(4*p)^2/2 := by |
| intro d |
| have hh := mul_nonneg hcoef (sq_nonneg (d-4*p)) |
| nlinarith only [hh] |
| constructor |
| · rintro _ ⟨d,rfl⟩ |
| change max 0 (max (p*(x.2+x.1*a*d)) (4*p*(x.2+x.1*a*d)-9*p))-x.1*a*d^2/2 ≤ _ |
| rw [← max_sub_sub_right,← max_sub_sub_right] |
| apply max_le |
| · exact (show 0-x.1*a*d^2/2 ≤ 0 by simpa using h0 d).trans (le_max_left _ _) |
| · apply max_le |
| · exact (h1 d).trans ((le_max_left _ _).trans (le_max_right _ _)) |
| · exact (h2 d).trans ((le_max_right _ _).trans (le_max_right _ _)) |
| · intro b hb |
| have he0 := hb (mem_range_self (0:ℝ)) |
| have he1 := hb (mem_range_self p) |
| have he2 := hb (mem_range_self (4*p)) |
| have hl0 : 0 ≤ quadraticCost (threeCore p) a x 0 := by |
| simp only [quadraticCost,shiftedField,mul_zero,add_zero,zero_pow (by decide : 2 ≠ 0),zero_div,sub_zero,threeCore] |
| exact le_max_left _ _ |
| have hl1 : p*x.2+x.1*a*p^2/2 ≤ quadraticCost (threeCore p) a x p := by |
| have hh := (le_max_left (p*(x.2+x.1*a*p)) (4*p*(x.2+x.1*a*p)-9*p)).trans (le_max_right (0:ℝ) _) |
| dsimp [quadraticCost,shiftedField,threeCore] |
| nlinarith only [hh] |
| have hl2 : 4*p*x.2-9*p+x.1*a*(4*p)^2/2 ≤ quadraticCost (threeCore p) a x (4*p) := by |
| have hh := (le_max_right (p*(x.2+x.1*a*(4*p))) (4*p*(x.2+x.1*a*(4*p))-9*p)).trans (le_max_right (0:ℝ) _) |
| dsimp [quadraticCost,shiftedField,threeCore] |
| nlinarith only [hh] |
| exact max_le (hl0.trans he0) (max_le (hl1.trans he1) (hl2.trans he2)) |
|
|
| lemma support_first_zero {f : ℝ → ℝ} {x w : Parameter} (hx : x ∈ pressureDomain) |
| (hw : SupportOn pressureDomain (fun y => f y.2) x w) : w.1=0 := by |
| change 0 < x.1 at hx |
| have h1 := hw (2*x.1,x.2) (by change 0 < 2*x.1; positivity) |
| have h2 := hw (x.1/2,x.2) (by change 0 < x.1/2; positivity) |
| dsimp [pDot] at h1 h2 |
| nlinarith [hx] |
|
|
| lemma firstMass_pos {p : ℝ} (hp : 0 < p) : 0 < firstMass p := by unfold firstMass; positivity |
| lemma firstMass_mul {p : ℝ} (hp : 0 < p) : firstMass p*p = 3/2 := by |
| unfold firstMass |
| field_simp |
|
|
| lemma threeMeanField_initial {p : ℝ} (hp : 0 < p) : threeMeanField p (firstMass p) initialPoint=0 := by |
| have hh := firstMass_mul hp |
| have hh2 : firstMass p*p^2=3*p/2 := by linear_combination p*hh |
| have h1 : p*(-3/4)+firstMass p*p^2/2=0 := by nlinarith only [hh2] |
| have h2 : 4*p*(-3/4)-9*p+firstMass p*(4*p)^2/2=0 := by nlinarith only [hh2] |
| simp only [threeMeanField,initialPoint,one_mul,h1,h2,max_self] |
|
|
| lemma three_active_initial {p : ℝ} (hp : 0 < p) : |
| activeLifts (threeCore p) (firstMass p) (threeMeanField p (firstMass p)) initialPoint ⊆ range (triangleVertex p) := by |
| intro w hw |
| have hβ : initialPoint ∈ pressureDomain := by norm_num [initialPoint,pressureDomain] |
| have he := support_first_zero (f := fun h => max 0 (max (p*h) (4*p*h-9*p))) (show shiftedField (firstMass p) initialPoint w.2 ∈ pressureDomain from hβ) hw.2 |
| have hc := hw.1 |
| rw [threeMeanField_initial hp] at hc |
| have ha := firstMass_pos hp |
| have hap := firstMass_mul hp |
| have hap2 : firstMass p*p^2=3*p/2 := by linear_combination p*hap |
| have hapd : firstMass p*p*w.2=3*w.2/2 := by linear_combination w.2*hap |
| change w.1-firstMass p*w.2^2/2=0 at he |
| simp only [quadraticCost,threeCore,shiftedField,initialPoint,one_mul] at hc |
| change max 0 (max (p*(-3/4+firstMass p*w.2)) (4*p*(-3/4+firstMass p*w.2)-9*p))-firstMass p*w.2^2/2=0 at hc |
| rcases le_total 0 (max (p*(-3/4+firstMass p*w.2)) (4*p*(-3/4+firstMass p*w.2)-9*p)) with h | h |
| · rw [max_eq_right h] at hc |
| rcases le_total (p*(-3/4+firstMass p*w.2)) (4*p*(-3/4+firstMass p*w.2)-9*p) with hh | hh |
| · rw [max_eq_right hh] at hc |
| have hd : w.2=4*p := by |
| have hh1 : firstMass p*(w.2-4*p)^2=0 := by nlinarith only [hc,hap2,hapd] |
| have := (mul_eq_zero.mp hh1).resolve_left ha.ne' |
| nlinarith [sq_nonneg (w.2-4*p)] |
| refine ⟨2,?_⟩ |
| ext <;> norm_num [triangleVertex,show (2:Fin 3) ≠ 0 by decide,show (2:Fin 3) ≠ 1 by decide] |
| · rw [hd] at he; nlinarith only [he,hap2] |
| · exact hd.symm |
| · rw [max_eq_left hh] at hc |
| have hd : w.2=p := by |
| have hh1 : firstMass p*(w.2-p)^2=0 := by nlinarith only [hc,hap2,hapd] |
| have := (mul_eq_zero.mp hh1).resolve_left ha.ne' |
| nlinarith [sq_nonneg (w.2-p)] |
| refine ⟨1,?_⟩ |
| ext <;> norm_num [triangleVertex] |
| · rw [hd] at he; nlinarith only [he,hap2] |
| · exact hd.symm |
| · rw [max_eq_left h] at hc |
| have hd : w.2=0 := by |
| have hm : firstMass p*w.2^2=0 := by linarith only [hc] |
| have hs := (mul_eq_zero.mp hm).resolve_left ha.ne' |
| nlinarith [sq_nonneg w.2] |
| refine ⟨0,?_⟩ |
| ext <;> norm_num [triangleVertex] |
| · rw [hd] at he; simpa using he.symm |
| · exact hd.symm |
|
|
| end ContinuumTemperature |
| end |
| end |
|
|
| end OAI |
|
|
