openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Criticality /ExpandingCharacterGrowth.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.Criticality.NormalClosureGrowth | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function | |
| open scoped Classical Pointwise | |
| namespace CriticalPercolation.SparseGrowth | |
| variable {A B : Type*} [Group A] [Group B] [DecidableEq A] [DecidableEq B] | |
| lemma binaryProducts_map (f : B →* A) (g : ℕ → B) (n : ℕ) : | |
| (binaryProducts g n).image f = binaryProducts (fun i => f (g i)) n := by | |
| induction n with | |
| | zero => simp [binaryProducts] | |
| | succ n ih => | |
| simp only [binaryProducts, Finset.image_union] | |
| rw [←ih] | |
| congr 1 | |
| simp only [Finset.image_image] | |
| apply Finset.image_congr | |
| intro x _ | |
| exact map_mul f (g n) x | |
| lemma binaryProducts_card_norm_bound | |
| (φ : A →* Multiplicative ℂ) (g : ℕ → A) (b q : ℝ) | |
| (hb : 0 < b) (hq : 3 ≤ q) | |
| (he : ∀ n, ‖Multiplicative.toAdd (φ (g n))‖ = b*q^n) : | |
| ∀ n, (binaryProducts g n).card=2^n ∧ | |
| ∀ a∈binaryProducts g n, | |
| 2*‖Multiplicative.toAdd (φ a)‖ ≤ b*(q^n-1) := by | |
| intro n | |
| induction n with | |
| | zero => | |
| constructor | |
| · simp [binaryProducts] | |
| · intro a ha | |
| have ha1 : a=1 := by simpa [binaryProducts] using ha | |
| simp [ha1] | |
| | succ n ih => | |
| have hdis : Disjoint (binaryProducts g n) | |
| ((binaryProducts g n).image (fun x => g n*x)) := by | |
| apply Finset.disjoint_left.mpr | |
| intro a ha ha' | |
| obtain ⟨c,hc,hca⟩ := Finset.mem_image.mp ha' | |
| have heq : Multiplicative.toAdd (φ (g n)) = | |
| Multiplicative.toAdd (φ a)-Multiplicative.toAdd (φ c) := by | |
| rw [←hca,map_mul] | |
| change Multiplicative.toAdd (φ (g n)) = | |
| (Multiplicative.toAdd (φ (g n))+Multiplicative.toAdd (φ c))- | |
| Multiplicative.toAdd (φ c) | |
| abel | |
| have hn := norm_sub_le (Multiplicative.toAdd (φ a)) (Multiplicative.toAdd (φ c)) | |
| rw [←heq,he n] at hn | |
| have haB := ih.2 a ha | |
| have hcB := ih.2 c hc | |
| nlinarith | |
| constructor | |
| · simp only [binaryProducts,Finset.card_union_of_disjoint hdis, | |
| Finset.card_image_of_injective _ (mul_right_injective (g n)),ih.1,pow_succ] | |
| omega | |
| · intro a ha | |
| obtain ha|ha := Finset.mem_union.mp ha | |
| · have haB := ih.2 a ha | |
| have hpow : q^n ≤ q^(n+1) := by | |
| rw [pow_succ] | |
| nlinarith [pow_nonneg (by linarith : 0≤q) n] | |
| exact haB.trans (mul_le_mul_of_nonneg_left (sub_le_sub_right hpow 1) hb.le) | |
| · obtain ⟨c,hc,rfl⟩ := Finset.mem_image.mp ha | |
| have hn := norm_add_le (Multiplicative.toAdd (φ (g n))) (Multiplicative.toAdd (φ c)) | |
| have hcB := ih.2 c hc | |
| rw [he n] at hn | |
| rw [map_mul] | |
| change 2*‖Multiplicative.toAdd (φ (g n))+Multiplicative.toAdd (φ c)‖≤_ | |
| rw [pow_succ] | |
| have hqn : 0≤q^n := pow_nonneg (by linarith) n | |
| have hprod : 3*(b*q^n) ≤ (b*q^n)*q := by | |
| calc | |
| 3*(b*q^n) = (b*q^n)*3 := mul_comm _ _ | |
| _ ≤ (b*q^n)*q := mul_le_mul_of_nonneg_left hq (mul_nonneg hb.le hqn) | |
| nlinarith | |
| theorem no_expanding_short_character (S : Finset A) (h1 : (1:A)∈S) | |
| (d : ℕ) (hg : HasDegree S d) (f : B →* A) (hf : Injective f) | |
| (φ : B →* Multiplicative ℂ) (g : ℕ → B) (b q : ℝ) | |
| (hb : 0<b) (hq : 3≤q) | |
| (he : ∀ n, ‖Multiplicative.toAdd (φ (g n))‖=b*q^n) | |
| (L : ℕ) (hL : 0<L) (hw : ∀ n, f (g n)∈S^(L*(n+1))) : False := by | |
| apply not_quadratic_exponential_lower S h1 d hg L hL | |
| intro n | |
| have hc := (binaryProducts_card_norm_bound φ g b q hb hq he n).1 | |
| calc | |
| 2^n = ((binaryProducts g n).image f).card := by | |
| rw [Finset.card_image_of_injective _ hf,hc] | |
| _ = (binaryProducts (fun i => f (g i)) n).card := by rw [binaryProducts_map] | |
| _ ≤ (S^(L*n^2)).card := | |
| Finset.card_le_card (binaryProducts_subset_pow S h1 (fun i => f (g i)) L hw n) | |
| theorem HasPolynomialDegree.norm_conj_eigencharacter_le_one {d : ℕ} | |
| (hg : HasPolynomialDegree A d) (H : Subgroup A) [H.Normal] | |
| (a : A) (φ : H →* Multiplicative ℂ) (ζ : ℂ) | |
| (heigen : ∀ x : H, Multiplicative.toAdd (φ ((MulAut.conjNormal a) x)) = | |
| ζ * Multiplicative.toAdd (φ x)) | |
| (b : H) (hb : Multiplicative.toAdd (φ b) ≠ 0) : ‖ζ‖≤1 := by | |
| by_contra! hl | |
| obtain ⟨k,hk⟩ := pow_unbounded_of_one_lt (3:ℝ) hl | |
| obtain ⟨S,h1,hs,hgen,hgrowth⟩ := (hasPolynomialDegree_iff d).mp hg | |
| obtain ⟨L,hL⟩ := finite_subset_word_ball S h1 hs hgen {a,a⁻¹,(b:A)} | |
| have haL : a∈S^L := hL (by simp) | |
| have haiL : a⁻¹∈S^L := hL (by simp) | |
| have hbL : (b:A)∈S^L := hL (by simp) | |
| have hepow : ∀ m : ℕ, ∀ x : H, | |
| Multiplicative.toAdd (φ ((MulAut.conjNormal (a^m)) x)) = | |
| ζ^m * Multiplicative.toAdd (φ x) := by | |
| intro m | |
| induction m with | |
| | zero => intro x; simp | |
| | succ m ih => | |
| intro x | |
| rw [pow_succ',map_mul,MulAut.mul_apply,heigen,ih,pow_succ'] | |
| ring | |
| let g : ℕ → H := fun n => (MulAut.conjNormal (a^(k*n))) b | |
| apply no_expanding_short_character S h1 d hgrowth H.subtype H.subtype_injective | |
| φ g ‖Multiplicative.toAdd (φ b)‖ (‖ζ‖^k) (norm_pos_iff.mpr hb) hk.le | |
| (fun n => ?_) (2*L*k+L+1) (by omega) (fun n => ?_) | |
| · dsimp only [g] | |
| rw [hepow,norm_mul,norm_pow,pow_mul,mul_comm] | |
| · change ((MulAut.conjNormal (a^(k*n))) b : A)∈_ | |
| rw [MulAut.conjNormal_apply] | |
| have haP : a^(k*n)∈S^(L*(k*n)) := by | |
| rw [pow_mul] | |
| exact Finset.pow_mem_pow haL | |
| have haiP : (a^(k*n))⁻¹∈S^(L*(k*n)) := by | |
| rw [←inv_pow,pow_mul] | |
| exact Finset.pow_mem_pow haiL | |
| apply Finset.pow_subset_pow_right h1 | |
| (show L*(k*n)+L+L*(k*n)≤(2*L*k+L+1)*(n+1) by nlinarith) | |
| rw [pow_add,pow_add] | |
| exact Finset.mul_mem_mul (Finset.mul_mem_mul haP hbL) haiP | |
| end CriticalPercolation.SparseGrowth | |
| end | |
| end | |
| end OAI | |