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f205175 | 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 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ActionOrbitTransport
namespace OAI
section
noncomputable section
open Set Function
open scoped Classical BigOperators
namespace CriticalPercolation.HomogeneousAnalysis
universe u v
variable {G : Type u} {X : Type v} [Group G] [MulAction G X]
open ActionOrbitTransport
lemma pair_fiber_eq_suborbit (x y : X) :
{w : X | pairOrbit G X x w=pairOrbit G X x y}=stabilizerOrbit G X x y := by
ext w
simp only [Set.mem_ofPred_eq]
rw [eq_comm,pairOrbit_eq,mem_stabilizerOrbit]
lemma reversed_pair_fiber_eq_suborbit (x y : X) :
{w : X | pairOrbit G X w y=pairOrbit G X x y}=stabilizerOrbit G X y x := by
ext w
simp only [Set.mem_ofPred_eq]
rw [eq_comm,pairOrbit_eq,mem_stabilizerOrbit]
exact exists_congr (fun g => and_comm)
variable (ht : ∀ x y : X, ∃ g : G, g • x=y)
(hf : ∀ x y : X, (stabilizerOrbit G X x y).Finite)
(hu : ∀ x y : X, suborbitCard G X x y=suborbitCard G X y x)
include ht hf in
lemma pair_row_finite (o b x : X) :
{y : X | pairOrbit G X x y=pairOrbit G X o b}.Finite := by
obtain ⟨g,rfl⟩ := ht o x
rw [←smul_pairOrbit G X g o b]
simpa only [pair_fiber_eq_suborbit] using hf (g • o) (g • b)
include ht in
lemma pair_row_card (o b x : X) :
{y : X | pairOrbit G X x y=pairOrbit G X o b}.ncard=suborbitCard G X o b := by
obtain ⟨g,rfl⟩ := ht o x
rw [←smul_pairOrbit G X g o b,pair_fiber_eq_suborbit]
exact smul_suborbitCard G X g o b
include ht hf in
lemma pair_column_finite (o b y : X) :
{x : X | pairOrbit G X x y=pairOrbit G X o b}.Finite := by
obtain ⟨g,rfl⟩ := ht b y
rw [←smul_pairOrbit G X g o b,reversed_pair_fiber_eq_suborbit]
exact hf (g • b) (g • o)
include ht hu in
lemma pair_column_card (o b y : X) :
{x : X | pairOrbit G X x y=pairOrbit G X o b}.ncard=suborbitCard G X o b := by
obtain ⟨g,rfl⟩ := ht b y
rw [←smul_pairOrbit G X g o b,reversed_pair_fiber_eq_suborbit]
change suborbitCard G X (g • b) (g • o)=_
rw [smul_suborbitCard,hu]
include ht hf hu in
lemma finite_row_sum_bound (o b : X) (F Y : Finset X) (f : X → ℝ)
(hn : ∀ y∈Y, 0≤f y)
(hY : ∀ x∈F, ∀ y, pairOrbit G X x y=pairOrbit G X o b → y∈Y) :
(∑ x∈F, ∑ y∈(pair_row_finite ht hf o b x).toFinset, f y) ≤
(suborbitCard G X o b : ℝ)*∑ y∈Y, f y := by
have hr (x : X) (hx : x∈F) :
∑ y∈(pair_row_finite ht hf o b x).toFinset, f y=
∑ y∈Y, if pairOrbit G X x y=pairOrbit G X o b then f y else 0 := by
rw [←Finset.sum_filter]
apply Finset.sum_congr
· ext y
simp only [Set.Finite.mem_toFinset,Set.mem_ofPred_eq,Finset.mem_filter]
exact ⟨fun h => ⟨hY x hx y h,h⟩,And.right⟩
· intros; rfl
rw [Finset.sum_congr rfl hr,Finset.sum_comm,Finset.mul_sum]
apply Finset.sum_le_sum
intro y hy
rw [←Finset.sum_filter,Finset.sum_const,nsmul_eq_mul]
apply mul_le_mul_of_nonneg_right _ (hn y hy)
have hsub : F.filter (fun x => pairOrbit G X x y=pairOrbit G X o b)⊆
(pair_column_finite ht hf o b y).toFinset := by
intro x hx
exact (pair_column_finite ht hf o b y).mem_toFinset.mpr (Finset.mem_filter.mp hx).2
have hc := Finset.card_le_card hsub
rw [←Set.ncard_eq_toFinset_card _ (pair_column_finite ht hf o b y), pair_column_card ht hu] at hc
exact_mod_cast hc
end CriticalPercolation.HomogeneousAnalysis
end
end
end OAI
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