openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Growth /HomogeneousDimension.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGeometry | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousCover | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousGrowth | |
| import OAI.Probability.CriticalPercolation.Growth.HomogeneousPolynomialSpace | |
| import OAI.Probability.CriticalPercolation.Harmonic.GramGrowthScales | |
| import OAI.Probability.CriticalPercolation.Harmonic.HomogeneousGramScales | |
| import OAI.Probability.CriticalPercolation.LinearGroups.MatrixLowGrowthSubspace | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Module | |
| open scoped Classical Pointwise BigOperators Matrix | |
| namespace CriticalPercolation.HomogeneousAnalysis.Geometry | |
| open HarmonicAnalysis | |
| variable {A ι : Type*} [Fintype ι] (W : Geometry A) | |
| lemma harmonic_functionCombination (v : ι → A → ℝ) (hv : ∀ i, W.IsHarmonic (v i)) | |
| (a : ι → ℝ) : W.IsHarmonic (functionCombination v a) := by | |
| rw [functionCombination_eq_linearCombination,Fintype.linearCombination_apply] | |
| change (∑ i, a i • v i)∈W.harmonicSpace | |
| exact W.harmonicSpace.sum_mem (fun i _ => W.harmonicSpace.smul_mem (a i) (hv i)) | |
| lemma scale_ratio_strict (K s L r R : ℕ) (hr : 1 ≤ r) | |
| (hscale : 10^(2*L)*r ≤ R) (hL : 8*K^4*s < 10^(4*L)) : | |
| 8*(K:ℝ)^4*(s:ℝ)*(r:ℝ)^2 < (R:ℝ)^2 := by | |
| have hr' : (0:ℝ) < r := by exact_mod_cast (show 0 < r by omega) | |
| have hs : (10:ℝ)^(2*L)*(r:ℝ) ≤ R := by exact_mod_cast hscale | |
| have hp := pow_le_pow_left₀ (by positivity : 0 ≤ (10:ℝ)^(2*L)*(r:ℝ)) hs 2 | |
| have hLp : 8*(K:ℝ)^4*(s:ℝ) < (10:ℝ)^(4*L) := by exact_mod_cast hL | |
| have hh := mul_lt_mul_of_pos_right hLp (sq_pos_of_pos hr') | |
| have he : ((10:ℝ)^(2*L)*(r:ℝ))^2=(10:ℝ)^(4*L)*(r:ℝ)^2 := by | |
| rw [mul_pow,←pow_mul (10:ℝ)] | |
| congr 2 | |
| ring | |
| rw [he] at hp | |
| exact hh.trans_le hp | |
| theorem harmonic_independent_card_bound (d k : ℕ) (hg : W.HasDegree d) | |
| (v : ι → A → ℝ) (hharm : ∀ i, W.IsHarmonic (v i)) | |
| (hpol : ∀ i, W.HasPolynomialGrowth k (v i)) (hli : LinearIndependent ℝ v) | |
| (L : ℕ) (hL : 0 < L) | |
| (hgap : 8*(10^(4*(2*d+2*k+1)))^4*W.degree < 10^(4*L)) : | |
| Fintype.card ι ≤ 2*(10^(4*(2*d+2*k+1)))^L := by | |
| by_cases hm : Fintype.card ι=0 | |
| · simp only [hm]; positivity | |
| have hm' : 0 < Fintype.card ι := Nat.pos_of_ne_zero hm | |
| let K : ℕ := 10^(4*(2*d+2*k+1)) | |
| have hK : 1 < (K:ℝ) := by | |
| have hh : 10 ≤ K := by | |
| change 10 ≤ 10^(4*(2*d+2*k+1)) | |
| simpa only [pow_one] using Nat.pow_le_pow_right (by omega : 0 < 10) | |
| (by omega : 1 ≤ 4*(2*d+2*k+1)) | |
| have hh' : (10:ℝ) ≤ K := by exact_mod_cast hh | |
| linarith | |
| obtain ⟨r,R,hr,hR,hrR,hscale,hvol,hdet,hpack,hPD⟩ := | |
| W.function_gram_paired_scales d k hg v hpol hli hm' L hL 1 | |
| have hdet' : (functionGram (W.ball (5*R)) v).det ≤ | |
| (K:ℝ)^(Fintype.card ι)*(functionGram (W.ball R) v).det := by | |
| simpa only [K,Nat.cast_pow,Nat.cast_ofNat] using hdet | |
| obtain ⟨V,hVdim,hVratio⟩ := exists_low_growth_subspace | |
| (functionGram (W.ball R) v) (functionGram (W.ball (5*R)) v) hPD | |
| (functionGram_hermitian _ _) (functionGram_quadratic_mono (W.ball_mono (by omega)) v) | |
| (K:ℝ) hK hdet' | |
| obtain ⟨J,hJR,hJ,hJcard,hcover⟩ := W.exists_word_ball_packing R r | |
| have hJbound : J.card ≤ K^L := | |
| le_of_mul_le_mul_right (hJcard.trans hpack) (W.ball_card_pos r) | |
| have hstrict := scale_ratio_strict K W.degree L r R hr hscale hgap | |
| let avg : V →ₗ[ℝ] (J → ℝ) := | |
| { toFun := fun a c => ∑ x∈W.around c.val (2*r), functionCombination v a.val x | |
| map_add' := by | |
| intro a b | |
| funext c | |
| simp only [Submodule.coe_add,functionCombination,Pi.add_apply,add_mul,Finset.sum_add_distrib] | |
| map_smul' := by | |
| intro a b | |
| funext c | |
| simp only [Submodule.coe_smul,functionCombination,Pi.smul_apply,smul_eq_mul, | |
| mul_assoc,←Finset.mul_sum,RingHom.id_apply] } | |
| have hInjective : Injective avg := by | |
| apply (LinearMap.ker_eq_bot).mp | |
| rw [LinearMap.ker_eq_bot'] | |
| intro a ha | |
| apply Subtype.ext | |
| by_contra hane | |
| have hane' : a.val≠0 := hane | |
| have hQ : 0 < squareMass (W.ball R) (functionCombination v a.val) := by | |
| have hh := (Matrix.posDef_iff_dotProduct_mulVec.mp hPD).2 hane' | |
| simpa only [star_trivial,←functionGram_quadratic,matrixQuadratic] using hh | |
| have havg : ∀ c∈J, (∑ x∈W.around c (2*r), functionCombination v a.val x)=0 := by | |
| intro c hc | |
| exact congrFun ha ⟨c,hc⟩ | |
| have hc := W.harmonic_cover_kernel_bound (W.balanced_suborbits d hg) R r K hR hrR J hJR hJ hcover hvol | |
| (functionCombination v a.val) (W.harmonic_functionCombination v hharm _) havg | |
| have hl := hVratio a.val a.property | |
| rw [functionGram_quadratic,functionGram_quadratic] at hl | |
| have hm := mul_le_mul_of_nonneg_left hl | |
| (by positivity : 0 ≤ 8*(K:ℝ)^2*(W.degree:ℝ)*(r:ℝ)^2) | |
| have hlow := mul_lt_mul_of_pos_right hstrict hQ | |
| nlinarith | |
| have hdim : Module.finrank ℝ V ≤ J.card := by | |
| have hh := LinearMap.finrank_le_finrank_of_injective hInjective | |
| simpa only [Module.finrank_fintype_fun_eq_card,Fintype.card_coe] using hh | |
| exact hVdim.trans (Nat.mul_le_mul_left 2 (hdim.trans hJbound)) | |
| omit [Fintype ι] in | |
| theorem polynomialHarmonicSpace_finiteDimensional (d k : ℕ) | |
| (hg : W.HasDegree d) : | |
| FiniteDimensional ℝ (W.polynomialHarmonicSpace k) := by | |
| let K : ℕ := 10^(4*(2*d+2*k+1)) | |
| let L : ℕ := 8*K^4*W.degree+1 | |
| have hL : 0 < L := by dsimp [L]; omega | |
| have hgap : 8*K^4*W.degree < 10^(4*L) := by | |
| have hpow : L < 10^L := L.lt_two_pow_self.trans_le (Nat.pow_le_pow_left (by omega) L) | |
| have hm : 10^L ≤ 10^(4*L) := Nat.pow_le_pow_right (by omega) (by omega) | |
| exact (by dsimp [L]; omega : 8*K^4*W.degree < L).trans (hpow.trans_le hm) | |
| let B := 2*K^L | |
| let H := W.polynomialHarmonicSpace k | |
| let b := Module.Free.chooseBasis ℝ H | |
| have hbound {n : ℕ} (w : Fin n → H) (hw : LinearIndependent ℝ w) : n ≤ B := by | |
| have hj : LinearIndependent ℝ (fun i => (w i:A → ℝ)) := | |
| hw.map' H.subtype (LinearMap.ker_eq_bot.mpr Subtype.val_injective) | |
| have hh := W.harmonic_independent_card_bound d k hg (fun i => (w i:A → ℝ)) | |
| (fun i => (w i).property.1) (fun i => (w i).property.2) hj L hL hgap | |
| simpa only [Fintype.card_fin] using hh | |
| have hfinite : Finite (Module.Free.ChooseBasisIndex ℝ H) := by | |
| by_contra h | |
| let : Infinite (Module.Free.ChooseBasisIndex ℝ H) := not_finite_iff_infinite.mp h | |
| let e : Fin (B+1) ↪ Module.Free.ChooseBasisIndex ℝ H := | |
| (Fin.valEmbedding).trans (Infinite.natEmbedding _) | |
| have hh := hbound (fun i => b (e i)) (b.linearIndependent.comp e e.injective) | |
| omega | |
| let := hfinite | |
| exact Module.Finite.of_basis b | |
| end CriticalPercolation.HomogeneousAnalysis.Geometry | |
| end | |
| end | |
| end OAI | |