File size: 4,436 Bytes
92416a2 | 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 | import OAI.Dynamics.ConditionalShuffle.SweepAverage
namespace OAI
noncomputable section
open scoped Classical
namespace Thorp.Conditional
lemma mean_perm_eval {α : Type*} [Fintype α] [DecidableEq α] (f : α → ℝ) (a : α) :
mean (fun g : Equiv.Perm α => f (g a)) = mean f := by
let : Nonempty α := ⟨a⟩
have he (b : α) : mean (fun g : Equiv.Perm α => f (g b)) =
mean (fun g : Equiv.Perm α => f (g a)) := by
have hh := mean_equiv (Equiv.mulRight (Equiv.swap a b)) (fun g : Equiv.Perm α => f (g a))
change mean (fun g : Equiv.Perm α => f ((g * Equiv.swap a b) a)) = _ at hh
simpa only [Equiv.Perm.mul_apply, Equiv.swap_apply_left] using hh
calc
_ = mean (fun b : α => mean (fun g : Equiv.Perm α => f (g b))) := by
simp only [he, mean_const]
_ = mean (fun g : Equiv.Perm α => mean (fun b : α => f (g b))) := mean_comm _
_ = mean (fun _ : Equiv.Perm α => mean f) :=
mean_congr (fun g => mean_equiv g f)
_ = _ := mean_const _
lemma mean_perm_cross_nonpos {α : Type*} [Fintype α] [DecidableEq α] (w : α → ℝ)
(hw : ∑ a, w a = 0) (a b : α) (hab : a ≠ b) :
mean (fun g : Equiv.Perm α => w (g a) * w (g b)) ≤ 0 := by
let C := mean (fun g : Equiv.Perm α => w (g a) * w (g b))
have hc (x : α) (hx : x ≠ a) :
mean (fun g : Equiv.Perm α => w (g a) * w (g x)) = C := by
have hh := mean_equiv (Equiv.mulRight (Equiv.swap b x))
(fun g : Equiv.Perm α => w (g a) * w (g b))
change mean (fun g : Equiv.Perm α => w ((g * Equiv.swap b x) a) *
w ((g * Equiv.swap b x) b)) = _ at hh
simpa only [Equiv.Perm.mul_apply,
Equiv.swap_apply_of_ne_of_ne hab hx.symm, Equiv.swap_apply_left] using hh
have hz : mean (fun g : Equiv.Perm α => ∑ x, w (g a) * w (g x)) = 0 := by
have hh (g : Equiv.Perm α) : (∑ x, w (g a) * w (g x)) = 0 := by
rw [← Finset.mul_sum, Equiv.sum_comp g w, hw, mul_zero]
simp only [hh, mean_zero]
rw [mean_sum, Fintype.sum_eq_add_sum_subtype_ne _ a] at hz
have hs : (∑ x : {x : α // x ≠ a}, mean (fun g : Equiv.Perm α => w (g a) * w (g x.val))) =
Fintype.card {x : α // x ≠ a} * C := by
calc
_ = ∑ _ : {x : α // x ≠ a}, C := Finset.sum_congr rfl (fun x _ => hc x.val x.property)
_ = _ := by simp
rw [hs] at hz
have hnonneg := mean_nonneg (fun g : Equiv.Perm α => mul_self_nonneg (w (g a)))
have hcard : (0 : ℝ) < Fintype.card {x : α // x ≠ a} := by
have hh : 0 < Fintype.card {x : α // x ≠ a} :=
Fintype.card_pos_iff.mpr ⟨⟨b,hab.symm⟩⟩
exact_mod_cast hh
change C ≤ 0
nlinarith
lemma mean_permuted_matrix_energy_le {α β : Type*} [Fintype α] [Fintype β] [DecidableEq α]
[Nonempty α] (w : α → ℝ) (hw : ∑ a, w a = 0) (K : α → β → ℝ)
(hK : ∀ a y, 0 ≤ K a y) :
mean (fun σ : Equiv.Perm α => ∑ y, (∑ a, w (σ a) * K a y)^2) ≤
mean (fun a => w a ^ 2) * ∑ a, ∑ y, K a y ^ 2 := by
have hterm (a b : α) (y : β) :
mean (fun σ : Equiv.Perm α => (w (σ a) * K a y) * (w (σ b) * K b y)) ≤
if b = a then mean (fun x => w x ^ 2) * K a y ^ 2 else 0 := by
by_cases h : b = a
· subst b
have hh (σ : Equiv.Perm α) : (w (σ a) * K a y) * (w (σ a) * K a y) =
(w (σ a))^2 * (K a y)^2 := by ring
rw [ite_eq_left rfl]
simp only [hh, mean_mul_const]
rw [mean_perm_eval (fun x => w x ^ 2) a]
· rw [ite_eq_right h]
have hh (σ : Equiv.Perm α) : (w (σ a) * K a y) * (w (σ b) * K b y) =
(w (σ a) * w (σ b)) * (K a y * K b y) := by ring
simp only [hh, mean_mul_const]
exact mul_nonpos_of_nonpos_of_nonneg (mean_perm_cross_nonpos w hw a b (fun ha => h ha.symm))
(mul_nonneg (hK a y) (hK b y))
calc
_ = ∑ y, ∑ a, ∑ b,
mean (fun σ : Equiv.Perm α => (w (σ a) * K a y) * (w (σ b) * K b y)) := by
simp only [sq, Finset.sum_mul, Finset.mul_sum, mean_sum]
apply Finset.sum_congr rfl; intro y _
exact Finset.sum_comm
_ ≤ ∑ y, ∑ a, ∑ b,
if b = a then mean (fun x => w x ^ 2) * K a y ^ 2 else 0 :=
Finset.sum_le_sum (fun y _ => Finset.sum_le_sum (fun a _ =>
Finset.sum_le_sum (fun b _ => hterm a b y)))
_ = _ := by
simp only [Finset.sum_ite_eq', Finset.mem_univ, ite_true]
rw [Finset.sum_comm]
simp only [← Finset.mul_sum]
end Thorp.Conditional
end
end OAI
|