openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /PeriodicSpectrum.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Polynomial Module | |
| open scoped Classical | |
| namespace CriticalPercolation.LinearGrowth | |
| lemma multiset_prod_dvd_pow_card {R : Type*} [CommMonoid R] (s : Multiset R) (a : R) | |
| (h : ∀ b∈s, b∣a) : s.prod∣a^s.card := by | |
| induction s using Multiset.induction_on with | |
| | empty => simp | |
| | cons b s ih => | |
| simp only [Multiset.prod_cons,Multiset.card_cons,pow_succ'] | |
| exact mul_dvd_mul (h b (Multiset.mem_cons_self _ _)) | |
| (ih (fun c hc => h c (Multiset.mem_cons_of_mem hc))) | |
| lemma nilpotent_aeval_of_vanishing_roots {K R : Type*} [Field K] [Ring R] [Algebra K R] | |
| (a : R) (hp : (minpoly K a).Monic) (hs : (minpoly K a).Splits) | |
| (q : K[X]) (hq : ∀ x ∈ (minpoly K a).roots, q.IsRoot x) : IsNilpotent (aeval a q) := by | |
| have hd : minpoly K a ∣ q^(minpoly K a).roots.card := by | |
| nth_rw 1 [hs.eq_prod_roots_of_monic hp] | |
| convert multiset_prod_dvd_pow_card ((minpoly K a).roots.map (fun x => X-C x)) q ?_ using 1 | |
| · simp only [Multiset.card_map] | |
| · intro b hb | |
| obtain ⟨x,hx,rfl⟩ := Multiset.mem_map.mp hb | |
| exact dvd_iff_isRoot.mpr (hq x hx) | |
| exact ⟨_,by rw [←map_pow]; exact minpoly.dvd_iff.mp hd⟩ | |
| lemma exists_scalar_unipotent_power {K V : Type*} [Field K] [IsAlgClosed K] | |
| [AddCommGroup V] [Module K V] [FiniteDimensional K V] | |
| (A : End K V) (a : K) (ha : a≠0) | |
| (hr : ∀ r, (minpoly K A).IsRoot r → ∃ m : ℕ, 0 < m ∧ (r/a)^m=1) : | |
| ∃ m : ℕ, 0 < m ∧ IsNilpotent (A^m-a^m • (1:End K V)) := by | |
| let s := (minpoly K A).roots.toFinset | |
| have he (r : s) : ∃ m : ℕ, 0 < m ∧ (r.val/a)^m=1 := | |
| hr r.val (isRoot_of_mem_roots (Multiset.mem_toFinset.mp r.property)) | |
| choose m hm hm1 using he | |
| let N := ∏ r : s, m r | |
| have hN : 0 < N := Finset.prod_pos (fun r _ => hm r) | |
| have hpow (r : s) : (r.val/a)^N=1 := by | |
| have hd : m r∣N := Finset.dvd_prod_of_mem m (Finset.mem_univ r) | |
| obtain ⟨k,hk⟩ := hd | |
| rw [hk,pow_mul,hm1 r,one_pow] | |
| refine ⟨N,hN,?_⟩ | |
| have hn := nilpotent_aeval_of_vanishing_roots A (minpoly.monic (Algebra.IsIntegral.isIntegral A)) | |
| (IsAlgClosed.splits (minpoly K A)) (X^N-C (a^N)) ?_ | |
| · simpa only [map_sub,map_pow,aeval_X,aeval_C,Algebra.algebraMap_eq_smul_one,_root_.smul_pow,one_pow] using hn | |
| · intro r hr' | |
| have hrN := hpow ⟨r,Multiset.mem_toFinset.mpr hr'⟩ | |
| have hep : r^N=a^N := by | |
| apply (div_eq_one_iff_eq (pow_ne_zero _ ha)).mp | |
| simpa only [div_pow] using hrN | |
| simp only [IsRoot,eval_sub,eval_pow,eval_X,eval_C,hep,sub_self] | |
| end CriticalPercolation.LinearGrowth | |
| end | |
| end | |
| end OAI | |