openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /LinearGroups /ArchimedeanDetection.lean
| import Mathlib | |
| import OAI.Probability.CriticalPercolation.LinearGroups.CountableFieldEmbedding | |
| namespace OAI | |
| section | |
| noncomputable section | |
| open Set Function Cardinal | |
| open scoped Classical | |
| namespace CriticalPercolation.Arithmetic | |
| lemma exists_transcendental_complex_norm_gt_one : ∃ z : ℂ, Transcendental ℚ z ∧ 1<‖z‖ := by | |
| have hex : ∃ y : ℝ, Transcendental ℚ y := by | |
| by_contra hn | |
| push Not at hn | |
| let : Algebra.IsAlgebraic ℚ ℝ := ⟨fun x => Classical.not_not.mp (hn x)⟩ | |
| have hh := Algebra.IsAlgebraic.cardinalMk_le_max ℚ ℝ | |
| have hq : #(ℚ)≤ℵ₀ := mk_le_aleph0 | |
| rw [max_eq_right hq] at hh | |
| exact (aleph0_lt_mk (α:=ℝ)).not_ge hh | |
| obtain ⟨y,hy⟩ := hex | |
| obtain ⟨n,hn⟩ := exists_nat_gt (‖y‖+2) | |
| have hpos : 1<y+(n:ℝ) := by | |
| have hab : -‖y‖≤y := by simpa only [Real.norm_eq_abs] using neg_abs_le y | |
| linarith | |
| have ht : Transcendental ℚ (y+(n:ℝ)) := by | |
| intro halg | |
| exact hy (by simpa only [add_sub_cancel_right] using halg.sub (isAlgebraic_natCast (R:=ℚ) n)) | |
| refine ⟨(y+(n:ℝ):ℝ),?_,?_⟩ | |
| · exact (transcendental_algebraMap_iff (R:=ℚ) (S:=ℝ) (A:=ℂ) (algebraMap ℝ ℂ).injective).mpr ht | |
| · simpa only [Complex.norm_real,Real.norm_eq_abs,abs_of_pos (lt_trans zero_lt_one hpos)] using hpos | |
| lemma exists_embedding_sending_transcendental {k K F : Type} [Field k] [Field K] [Field F] | |
| [Algebra k K] [Algebra k F] [Countable k] [Countable K] [IsAlgClosed F] [Uncountable F] | |
| (x : K) (hx : Transcendental k x) (y : F) (hy : Transcendental k y) : | |
| ∃ f : K →+* F, f x=y := by | |
| let v : Unit → K := fun _ => x | |
| let w : Unit → F := fun _ => y | |
| have hv : AlgebraicIndependent k v := algebraicIndependent_unique_type_iff.mpr hx | |
| have hw : AlgebraicIndependent k w := algebraicIndependent_unique_type_iff.mpr hy | |
| let A := IntermediateField.adjoin k (Set.range v) | |
| let φ : A →+* F := (IsFractionRing.lift | |
| (algebraicIndependent_iff_injective_aeval.mp hw)).comp hv.reprField.toRingHom | |
| let a : A := ⟨x,IntermediateField.subset_adjoin k (Set.range v) ⟨(),rfl⟩⟩ | |
| have hφ : φ a=y := by | |
| let q : FractionRing (MvPolynomial Unit k) := algebraMap (MvPolynomial Unit k) _ (MvPolynomial.X ()) | |
| have ha : hv.aevalEquivField q=a := by | |
| apply Subtype.ext | |
| rw [hv.aevalEquivField_algebraMap_apply_coe] | |
| simp only [MvPolynomial.aeval_X] | |
| rfl | |
| rw [←ha] | |
| change IsFractionRing.lift (algebraicIndependent_iff_injective_aeval.mp hw) | |
| (hv.aevalEquivField.symm (hv.aevalEquivField q))=y | |
| rw [hv.aevalEquivField.symm_apply_apply] | |
| simp only [q,IsFractionRing.lift_algebraMap] | |
| exact MvPolynomial.aeval_X w () | |
| obtain ⟨f,hf⟩ := exists_extension_of_countable (k:=A) (K:=K) φ | |
| refine ⟨f,?_⟩ | |
| exact (RingHom.congr_fun hf a).trans hφ | |
| lemma exists_complex_embedding_norm_gt_one {K : Type} [Field K] [CharZero K] [Countable K] | |
| (x : K) (hx : Transcendental ℚ x) : ∃ f : K →+* ℂ, 1<‖f x‖ := by | |
| let : Uncountable ℂ := uncountable_complete_normed_field ℂ | |
| obtain ⟨y,hy,hyNorm⟩ := exists_transcendental_complex_norm_gt_one | |
| obtain ⟨f,hf⟩ := exists_embedding_sending_transcendental x hx y hy | |
| exact ⟨f,hf ▸ hyNorm⟩ | |
| end CriticalPercolation.Arithmetic | |
| end | |
| end | |
| end OAI | |