openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Growth /HomogeneousTransport.lean
| 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 | |