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135052a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 | import Mathlib
import OAI.Probability.SKBarriers.Replicas.ReplicaSampling
import OAI.Probability.SKBarriers.SpinGlass.FiniteSampling
namespace OAI
section
noncomputable section
open scoped BigOperators
open Classical
namespace SK.Analytic
namespace FiniteLaw
theorem cover_mixed_mass {X Y : Type*} [Fintype X] [Fintype Y]
(P : FiniteLaw X) (Q : FiniteLaw Y) (R : X → Y → Prop) (E : X → Prop)
{a δ : ℝ} (ha : 0<a) (hcov : P.prob (fun x => Q.prob (R x)<a)≤δ) :
P.prob E≤δ+Q.expect (fun y => P.prob (fun x => E x ∧ R x y))/a := by
have H := P.expect_mono (f:=fun x => a*(if E x then 1 else 0))
(g:=fun x => a*(if Q.prob (R x)<a then 1 else 0)+Q.prob (fun y => E x ∧ R x y)) (fun x => by
by_cases hx : E x
· simp only [hx,true_and,ite_true,mul_one]
by_cases hc : Q.prob (R x)<a
· simp only [ite_eq_left hc,mul_one]
linarith only [Q.prob_nonneg (R x)]
· simp only [ite_eq_right hc,mul_zero,zero_add]
exact le_of_not_gt hc
· simp only [hx,false_and,ite_false,mul_zero,prob_false,add_zero]
split_ifs <;> simp only [mul_one,mul_zero] <;> linarith only [ha])
rw [expect_const_mul,expect_add,expect_const_mul] at H
simp only [expect_indicator_eq_prob] at H
have he : P.expect (fun x => Q.prob (fun y => E x ∧ R x y))=
Q.expect (fun y => P.prob (fun x => E x ∧ R x y)) := P.expect_expect_comm Q _
rw [he] at H
have HC := mul_le_mul_of_nonneg_left hcov ha.le
apply (mul_le_mul_iff_right₀ ha).mp
rw [mul_add,mul_div_cancel₀ _ ha.ne']
nlinarith only [H,HC]
theorem expect_involution {X : Type*} [Fintype X] (P : FiniteLaw X)
(T : X → X) (hT : Function.Involutive T) (hP : ∀x,P.weight (T x)=P.weight x) (f : X → ℝ) :
P.expect (fun x => f (T x))=P.expect f := by
let e : X ≃ X := ⟨T,T,hT,hT⟩
apply Fintype.sum_equiv e
intro x
change P.weight x*f (T x)=P.weight (T x)*f (T x)
rw [hP]
theorem iid_pair_expect {X : Type*} [Fintype X] (P : FiniteLaw X) (f : X → X → ℝ) :
(P.iid (Fin 2)).expect (fun s => f (s 0) (s 1))=P.expect (fun x => P.expect (f x)) := by
rw [iid_expect_fin_succ P 1]
change (P.iid (Fin 1)).expect (fun s => P.expect (fun y => f (s 0) y))=P.expect (fun x => P.expect (f x))
exact iid_expect_single P (0:Fin 1) (fun x => P.expect (f x))
end FiniteLaw
theorem gibbs_prob_eq_sum {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop) :
(gibbsFiniteLaw β J).prob E=∑x,if E x then gibbs β J x else 0 := by
simp only [FiniteLaw.prob,FiniteLaw.expect,gibbsFiniteLaw,mul_ite,mul_one,mul_zero]
theorem gibbs_expect_flip {n : ℕ} (β : ℝ) (J : Disorder n) (f : Config n → ℝ) :
(gibbsFiniteLaw β J).expect (fun v => f (flip v))=(gibbsFiniteLaw β J).expect f :=
(gibbsFiniteLaw β J).expect_involution flip flip_flip (gibbs_flip β J) f
theorem mixed_abs_neighbor_le_signed {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop)
{q a₁ : ℝ} (haq : a₁<q) (v : Config n) :
(gibbsFiniteLaw β J).prob (fun x => E x ∧ q≤|overlap x v|)≤
(gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ E x)+
(gibbsFiniteLaw β J).prob (fun x => a₁<overlap (flip v) x ∧ E x) := by
refine ((gibbsFiniteLaw β J).prob_mono (fun x hx => ?_)).trans ((gibbsFiniteLaw β J).prob_or _ _)
rw [overlap_comm] at hx
rcases le_abs.mp hx.2 with h|h
· exact Or.inl ⟨haq.trans_le h,hx.1⟩
· apply Or.inr
refine ⟨?_,hx.1⟩
rw [overlap_comm,overlap_flip_right,overlap_comm]
linarith only [haq,h]
theorem covered_mixed_mass_le_signed_average {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop)
{q a₁ a δ : ℝ} (haq : a₁<q) (ha : 0<a)
(hcov : (gibbsFiniteLaw β J).prob (fun x => (gibbsFiniteLaw β J).prob (fun y => q≤|overlap x y|)<a)≤δ) :
(gibbsFiniteLaw β J).prob E≤δ+(2/a)*(gibbsFiniteLaw β J).expect (fun v =>
(gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ E x)) := by
have H := (gibbsFiniteLaw β J).cover_mixed_mass (gibbsFiniteLaw β J) (fun x y => q≤|overlap x y|) E ha hcov
have HB := (gibbsFiniteLaw β J).expect_mono (fun v => mixed_abs_neighbor_le_signed β J E haq v)
rw [FiniteLaw.expect_add] at HB
have HF := gibbs_expect_flip β J (fun v => (gibbsFiniteLaw β J).prob (fun x => a₁<overlap v x ∧ E x))
rw [HF] at HB
refine H.trans ?_
apply add_le_add_right
have HD := div_le_div_of_nonneg_right HB ha.le
refine HD.trans_eq ?_
ring
def pairStripSet (n : ℕ) (a₀ a₁ : ℝ) : Finset (ReplicaConfig n 2) :=
Finset.univ.filter (fun s => a₀≤overlap (s 0) (s 1) ∧ overlap (s 0) (s 1)≤a₁)
theorem pairStrip_expect {n : ℕ} (β : ℝ) (J : Disorder n) (a₀ a₁ : ℝ) :
(gibbsFiniteLaw β J).expect (fun v => (gibbsFiniteLaw β J).prob (fun x => a₀≤overlap v x ∧ overlap v x≤a₁))=
replicaGibbsMass β J (pairStripSet n a₀ a₁) := by
rw [← iid_gibbs_replica β J (pairStripSet n a₀ a₁)]
simp only [pairStripSet,Finset.mem_filter,Finset.mem_univ,true_and,FiniteLaw.prob]
convert ((gibbsFiniteLaw β J).iid_pair_expect (fun v x => if a₀≤overlap v x ∧ overlap v x≤a₁ then 1 else 0)).symm using 1
all_goals congr 1
· funext v
congr 1
funext x
split_ifs <;> rfl
end SK.Analytic
end
end
end OAI
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