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import OAI.Probability.CriticalPercolation.ProductMeasure.Prob
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteProductPin
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
open scoped Classical
namespace CriticalPercolation.FiniteVariance
lemma sq_div_le_self {a r : ℝ} (ha : 0 ≤ a) (hr : 0 ≤ r) (hra : r ≤ a) :
r^2/a ≤ r := by
rcases eq_or_lt_of_le ha with hz | hp
· have : r = 0 := by linarith
simp [this]
· apply (div_le_iff₀ hp).mpr
nlinarith
lemma binary_split {a b x y : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b)
(hx : 0 ≤ x) (hy : 0 ≤ y) (hxa : x ≤ a) (hyb : y ≤ b) :
x^2/a + y^2/b - (x+y)^2/(a+b) = a*b/(a+b)*(y/b-x/a)^2 := by
rcases eq_or_lt_of_le ha with hz | hp
· have hx0 : x = 0 := by linarith
simp [← hz, hx0]
· rcases eq_or_lt_of_le hb with hz | hq
· have hy0 : y = 0 := by linarith
simp [← hz, hy0]
· have hab : a+b ≠ 0 := (add_pos hp hq).ne'
field_simp
ring
end CriticalPercolation.FiniteVariance
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E : Type*} [Fintype E]
lemma expect_prop_mask_pin (w : E → Prop → ℝ) (e : E) (b : Prop) (f : (E → Prop) → ℝ) :
expect w (fun ω => if ω e = b then f ω else 0) = w e b*expect (pin w e b) f := by
convert expect_mask_pin w e b f using 1
congr 1
funext ω
by_cases h : ω e = b <;> simp [h]
lemma expect_split_pin (w : E → Prop → ℝ) (e : E) (f : (E → Prop) → ℝ) :
expect w f = w e False*expect (pin w e False) f +
w e True*expect (pin w e True) f := by
calc
expect w f = expect w (fun ω => (if ω e = False then f ω else 0) +
(if ω e = True then f ω else 0)) := by
congr 1
funext ω
by_cases h : ω e <;> simp [h]
_ = _ := by
rw [expect_add]
apply congrArg₂ (· + ·)
· convert expect_prop_mask_pin w e False f using 1
congr 1
funext ω
by_cases h : ω e <;> simp [h]
· convert expect_prop_mask_pin w e True f using 1
congr 1
funext ω
by_cases h : ω e <;> simp [h]
lemma expect_congr_support (w : E → Prop → ℝ) (f g : Set E → ℝ)
(h : ∀ ω, density w ω ≠ 0 → f ω = g ω) : expect w f = expect w g := by
unfold expect
apply Finset.sum_congr rfl
intro ω _
by_cases hd : density w ω = 0
· simp [hd]
· rw [h ω hd]
lemma density_pin_support (w : E → Prop → ℝ) (e : E) (b : Prop) (ω : E → Prop)
(h : density (pin w e b) ω ≠ 0) : ω e = b := by
rw [density_pin] at h
by_contra he
simp [he] at h
lemma density_pin_old_support (w : E → Prop → ℝ) (e : E) (b : Prop) (ω : E → Prop)
(hw : w e b ≠ 0) (h : density (pin w e b) ω ≠ 0) : density w ω ≠ 0 := by
have hbit := density_pin_support w e b ω h
rw [density,← Finset.prod_erase_mul _ _ (Finset.mem_univ e),hbit]
rw [density_pin,ite_eq_left hbit,one_mul] at h
exact mul_ne_zero h hw
lemma expect_pin_mask (w : E → Prop → ℝ) (e : E) (b : Prop) (f : Set E → ℝ) :
expect (pin w e b) (fun ω => if ω e = b then f ω else 0) =
expect (pin w e b) f := by
apply expect_congr_support
intro ω h
rw [ite_eq_left (density_pin_support w e b ω h)]
noncomputable def potential (w : E → Prop → ℝ) (U R : Set E → Prop) : ℝ :=
prob w R - (prob w R)^2/prob w U
lemma potential_le (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b) (U R : Set E → Prop) :
potential w U R ≤ prob w R := by
have hU := prob_nonneg w hp U
dsimp [potential]
exact sub_le_self _ (div_nonneg (sq_nonneg _) hU)
lemma potential_nonneg (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b)
(U R : Set E → Prop) (hRU : ∀ ω, R ω → U ω) : 0 ≤ potential w U R := by
dsimp [potential]
exact sub_nonneg.mpr (FiniteVariance.sq_div_le_self (prob_nonneg w hp U)
(prob_nonneg w hp R) (prob_mono w hp hRU))
lemma potential_split (w : E → Prop → ℝ) (hp : ∀ e b, 0 ≤ w e b)
(e : E) (U R : Set E → Prop) (hRU : ∀ ω, R ω → U ω)
{p : ℝ} (hp0 : 0 < p) (hp1 : p < 1) (hw0 : w e False = 1-p) (hw1 : w e True = p) :
let U0 := prob (pin w e False) U
let U1 := prob (pin w e True) U
let R0 := prob (pin w e False) R
let R1 := prob (pin w e True) R
potential w U R - ((1-p)*potential (pin w e False) U R +
p*potential (pin w e True) U R) =
p*(1-p)*U0*U1/((1-p)*U0+p*U1)*(R1/U1-R0/U0)^2 := by
dsimp only
have hUn := fun b => prob_nonneg (pin w e b) (pin_nonneg w hp e b) U
have hRn := fun b => prob_nonneg (pin w e b) (pin_nonneg w hp e b) R
have hRU' := fun b => prob_mono (pin w e b) (pin_nonneg w hp e b) hRU
have hU : prob w U = (1-p)*prob (pin w e False) U+p*prob (pin w e True) U := by
have hh := expect_split_pin w e (fun ω => if U ω then 1 else 0)
simpa only [prob,hw0,hw1] using hh
have hR : prob w R = (1-p)*prob (pin w e False) R+p*prob (pin w e True) R := by
have hh := expect_split_pin w e (fun ω => if R ω then 1 else 0)
simpa only [prob,hw0,hw1] using hh
have hq : 0 < 1-p := sub_pos.mpr hp1
have hb := FiniteVariance.binary_split
(mul_nonneg hq.le (hUn False)) (mul_nonneg hp0.le (hUn True))
(mul_nonneg hq.le (hRn False)) (mul_nonneg hp0.le (hRn True))
(mul_le_mul_of_nonneg_left (hRU' False) hq.le)
(mul_le_mul_of_nonneg_left (hRU' True) hp0.le)
have hscale : ∀ (c d : ℝ), (c*d)^2/c = c*d^2 := by
intro c d
by_cases hc : c = 0
· simp [hc]
· field_simp
simp only [mul_div_mul_left _ _ hq.ne',mul_div_mul_left _ _ hp0.ne'] at hb
have hx : ∀ (c d u : ℝ), (c*d)^2/(c*u) = c*d^2/u := by
intro c d u
rw [div_mul_eq_div_div,hscale]
simp only [hx] at hb
dsimp [potential]
rw [hU,hR]
convert hb using 1 <;> ring
end CriticalPercolation.FiniteProduct
end
end
end OAI
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