openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Growth /HomogeneousPolynomialSpace.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open scoped Classical Pointwise BigOperators | |
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry | |
| variable {A : Type*} (W : Geometry A) | |
| def harmonicSpace : Submodule ℝ (A → ℝ) where | |
| carrier := {u | W.IsHarmonic u} | |
| zero_mem' := by intro x; simp | |
| add_mem' := by | |
| intro u v hu hv x | |
| change (∑ s∈W.neighbors x, ((u x+v x)-(u s+v s)))=0 | |
| simp_rw [show ∀ s:A, (u x+v x)-(u s+v s)=(u x-u s)+(v x-v s) by intro s; ring] | |
| rw [Finset.sum_add_distrib,hu x,hv x,add_zero] | |
| smul_mem' := by | |
| intro c u hu x | |
| change (∑ s∈W.neighbors x, (c*u x-c*u s))=0 | |
| simp_rw [←mul_sub] | |
| rw [←Finset.mul_sum,hu x,mul_zero] | |
| def HasPolynomialGrowth (k : ℕ) (u : A → ℝ) : Prop := | |
| ∃ C : ℝ, 0 ≤ C ∧ ∀ n, ∀ x∈W.ball n, |u x| ≤ C*((n:ℝ)+1)^k | |
| def polynomialGrowthSpace (k : ℕ) : Submodule ℝ (A → ℝ) where | |
| carrier := {u | W.HasPolynomialGrowth k u} | |
| zero_mem' := ⟨0,le_rfl,by simp⟩ | |
| add_mem' := by | |
| rintro u v ⟨C,hC,hu⟩ ⟨D,hD,hv⟩ | |
| refine ⟨C+D,add_nonneg hC hD,?_⟩ | |
| intro n x hx | |
| change |u x+v x| ≤ _ | |
| calc | |
| _ ≤ |u x|+|v x| := abs_add_le _ _ | |
| _ ≤ C*((n:ℝ)+1)^k+D*((n:ℝ)+1)^k := add_le_add (hu n x hx) (hv n x hx) | |
| _ = _ := by ring | |
| smul_mem' := by | |
| rintro c u ⟨C,hC,hu⟩ | |
| refine ⟨|c| * C,mul_nonneg (abs_nonneg _) hC,?_⟩ | |
| intro n x hx | |
| change |c*u x| ≤ _ | |
| rw [abs_mul,mul_assoc] | |
| exact mul_le_mul_of_nonneg_left (hu n x hx) (abs_nonneg _) | |
| def polynomialHarmonicSpace (k : ℕ) : Submodule ℝ (A → ℝ) := | |
| W.harmonicSpace ⊓ W.polynomialGrowthSpace k | |
| lemma mem_polynomialHarmonicSpace (k : ℕ) (u : A → ℝ) : | |
| u∈W.polynomialHarmonicSpace k ↔ W.IsHarmonic u ∧ W.HasPolynomialGrowth k u := Iff.rfl | |
| lemma polynomial_growth_iff_length (k : ℕ) (u : A → ℝ) : | |
| W.HasPolynomialGrowth k u ↔ ∃ C : ℝ, 0 ≤ C ∧ | |
| ∀ x, |u x| ≤ C*((W.length x:ℝ)+1)^k := by | |
| constructor | |
| · rintro ⟨C,hC,hu⟩ | |
| exact ⟨C,hC,fun x => hu _ _ (W.mem_ball_length x)⟩ | |
| · rintro ⟨C,hC,hu⟩ | |
| refine ⟨C,hC,?_⟩ | |
| intro n x hx | |
| apply (hu x).trans | |
| apply mul_le_mul_of_nonneg_left _ hC | |
| apply pow_le_pow_left₀ (by positivity) | |
| exact_mod_cast Nat.add_le_add_right (W.length_le_iff.mpr hx) 1 | |
| lemma finite_family_growth_bound {ι : Type*} [Fintype ι] (k : ℕ) (v : ι → A → ℝ) | |
| (hv : ∀ i, W.HasPolynomialGrowth k (v i)) : | |
| ∃ M : ℝ, 0 ≤ M ∧ ∀ n, 1 ≤ n → ∀ i, ∀ x∈W.ball n, | |
| |v i x| ≤ M*(n:ℝ)^k := by | |
| choose C hC hCv using hv | |
| let M := (∑ i, C i)*2^k | |
| have hm0 : 0 ≤ M := mul_nonneg (Finset.sum_nonneg (fun i _ => hC i)) (by positivity) | |
| refine ⟨M,hm0,?_⟩ | |
| intro n hn i x hx | |
| have hn' : (1:ℝ) ≤ n := by exact_mod_cast hn | |
| have hCi : C i ≤ ∑ j, C j := Finset.single_le_sum (fun j _ => hC j) (Finset.mem_univ i) | |
| calc | |
| _ ≤ C i*((n:ℝ)+1)^k := hCv i n x hx | |
| _ ≤ (∑ j, C j)*((n:ℝ)+1)^k := mul_le_mul_of_nonneg_right hCi (by positivity) | |
| _ ≤ (∑ j, C j)*(2*(n:ℝ))^k := mul_le_mul_of_nonneg_left | |
| (pow_le_pow_left₀ (by positivity) (by linarith) k) (Finset.sum_nonneg (fun j _ => hC j)) | |
| _ = M*(n:ℝ)^k := by dsimp [M]; rw [mul_pow]; ring | |
| end CriticalPercolation.HomogeneousAnalysis.Geometry | |
| end | |
| end | |
| end OAI | |