openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Arithmetic /ConcurrentDenominators.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.Arithmetic.PolygonArithmetic | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section ConcurrentDenominators | |
| noncomputable def lineNormal (a : ℕ) (j : Fin 4) : CutRing × CutRing := | |
| ![(1,0),(0,1),(-(cutTau^a),1),(1,-(cutTau^a))] j | |
| noncomputable def lineDet (a : ℕ) (i j : Fin 4) : CutRing := | |
| (lineNormal a i).1*(lineNormal a j).2-(lineNormal a j).1*(lineNormal a i).2 | |
| theorem cutForm_normal (a : ℕ) (j : Fin 4) (z : ℝ × ℝ) : | |
| cutForm a j z = ordinary (lineNormal a j).1*z.1+ordinary (lineNormal a j).2*z.2 := by | |
| fin_cases j <;> simp [cutForm,lineNormal] <;> ring | |
| theorem lineDet_ne_zero {a : ℕ} (ha : 0<a) (i j : Fin 4) (hij : i ≠ j) : | |
| lineDet a i j ≠ 0 := by | |
| have hp : 1 < ordinary (cutTau^a) := by | |
| simpa using one_lt_pow₀ Real.one_lt_goldenRatio (show a≠0 by omega) | |
| have hn : cutTau^a ≠ 0 := by | |
| intro he; have hh := congrArg ordinary he; simp only [map_zero] at hh; linarith | |
| have hm : 1-cutTau^a*cutTau^a ≠ 0 := by | |
| intro he; have hh := congrArg ordinary he | |
| simp only [map_sub,map_one,map_mul,map_zero] at hh | |
| nlinarith | |
| have hm' : cutTau^a*cutTau^a-1 ≠ 0 := by | |
| intro he; apply hm; linear_combination -he | |
| fin_cases i <;> fin_cases j <;> | |
| simp_all [lineDet,lineNormal] | |
| theorem intersection_pair_denominator (a : ℕ) (i j : Fin 4) | |
| (c d : CutRing) (z : ℝ × ℝ) | |
| (hi : cutForm a i z = ordinary c) (hj : cutForm a j z = ordinary d) : | |
| ∃ u v : CutRing, | |
| ordinary u=((QuadraticAlgebra.norm (lineDet a i j) : ℤ) : ℝ)*z.1 ∧ | |
| ordinary v=((QuadraticAlgebra.norm (lineDet a i j) : ℤ) : ℝ)*z.2 := by | |
| let δ := lineDet a i j | |
| let x := c*(lineNormal a j).2-d*(lineNormal a i).2 | |
| let y := (lineNormal a i).1*d-(lineNormal a j).1*c | |
| have hx : ordinary x=ordinary δ*z.1 := by | |
| rw [cutForm_normal] at hi hj | |
| simp only [x,δ,lineDet,map_sub,map_mul] | |
| linear_combination -(ordinary (lineNormal a j).2)*hi+(ordinary (lineNormal a i).2)*hj | |
| have hy : ordinary y=ordinary δ*z.2 := by | |
| rw [cutForm_normal] at hi hj | |
| simp only [y,δ,lineDet,map_sub,map_mul] | |
| linear_combination -(ordinary (lineNormal a i).1)*hj+(ordinary (lineNormal a j).1)*hi | |
| refine ⟨x*star δ,y*star δ,?_,?_⟩ <;> | |
| simp only [map_mul,hx,hy,← conjugate_eq_ordinary_star] | |
| · rw [← cut_norm_identity δ]; ring | |
| · rw [← cut_norm_identity δ]; ring | |
| noncomputable def commonVertexDenominator (a : ℕ) : ℕ := | |
| (∏ i : Fin 4, ∏ j : Fin 4, | |
| if i=j then (1 : ℤ) else QuadraticAlgebra.norm (lineDet a i j)).natAbs | |
| theorem commonVertexDenominator_pos {a : ℕ} (ha : 0<a) : 0<commonVertexDenominator a := by | |
| apply Int.natAbs_pos.mpr | |
| apply Finset.prod_ne_zero_iff.mpr | |
| intro i _ | |
| apply Finset.prod_ne_zero_iff.mpr | |
| intro j _ | |
| split_ifs with h | |
| · norm_num | |
| · exact cut_norm_nonzero (lineDet_ne_zero ha i j h) | |
| theorem pair_norm_dvd_vertexDenominator (a : ℕ) (i j : Fin 4) (hij : i≠j) : | |
| QuadraticAlgebra.norm (lineDet a i j) ∣ (commonVertexDenominator a : ℤ) := by | |
| apply Int.dvd_natAbs.mpr | |
| have hi := Finset.dvd_prod_of_mem | |
| (fun i : Fin 4 => ∏ j : Fin 4, if i=j then (1 : ℤ) else QuadraticAlgebra.norm (lineDet a i j)) | |
| (Finset.mem_univ i) | |
| have hj := Finset.dvd_prod_of_mem | |
| (fun j : Fin 4 => if i=j then (1 : ℤ) else QuadraticAlgebra.norm (lineDet a i j)) | |
| (Finset.mem_univ j) | |
| have hh : QuadraticAlgebra.norm (lineDet a i j) ∣ | |
| ∏ j : Fin 4, if i=j then (1 : ℤ) else QuadraticAlgebra.norm (lineDet a i j) := by | |
| simpa only [ite_eq_right hij] using hj | |
| exact hh.trans hi | |
| theorem all_intersections_common_denominator {a : ℕ} (_ha : 0<a) | |
| (i j : Fin 4) (hij : i≠j) (c d : CutRing) (z : ℝ × ℝ) | |
| (hi : cutForm a i z = ordinary c) (hj : cutForm a j z = ordinary d) : | |
| ∃ u v : CutRing, | |
| ordinary u=(commonVertexDenominator a : ℝ)*z.1 ∧ | |
| ordinary v=(commonVertexDenominator a : ℝ)*z.2 := by | |
| obtain ⟨u,v,hu,hv⟩ := intersection_pair_denominator a i j c d z hi hj | |
| obtain ⟨k,hk⟩ := pair_norm_dvd_vertexDenominator a i j hij | |
| have hkr : (commonVertexDenominator a : ℝ)= | |
| ((QuadraticAlgebra.norm (lineDet a i j) : ℤ) : ℝ)*(k : ℝ) := by exact_mod_cast hk | |
| refine ⟨(k : CutRing)*u,(k : CutRing)*v,?_,?_⟩ <;> | |
| simp only [map_mul,map_intCast,hu,hv,hkr] <;> ring | |
| end ConcurrentDenominators | |
| section VertexResidues | |
| noncomputable def cutResidue (D : ℕ) (z : CutRing) : CutRing := | |
| ⟨z.re % (D : ℤ),z.im % (D : ℤ)⟩ | |
| noncomputable def cutQuotient (D : ℕ) (z : CutRing) : CutRing := | |
| ⟨z.re / (D : ℤ),z.im / (D : ℤ)⟩ | |
| theorem cut_residue_decomposition (D : ℕ) (z : CutRing) : | |
| z = cutResidue D z + (D : CutRing)*cutQuotient D z := by | |
| ext <;> simp [cutResidue,cutQuotient,Int.emod_add_mul_ediv] | |
| noncomputable def finiteCutResidue (D : ℕ) (t : Fin D × Fin D) : CutRing := | |
| ⟨t.1.val,t.2.val⟩ | |
| theorem cutResidue_exists_finite {D : ℕ} (hD : 0<D) (z : CutRing) : | |
| ∃ t : Fin D × Fin D, cutResidue D z=finiteCutResidue D t := by | |
| have hDi : (0 : ℤ)<D := by exact_mod_cast hD | |
| let p : Fin D := ⟨(z.re % (D : ℤ)).toNat,by | |
| have h := Int.emod_lt_of_pos z.re hDi | |
| have hn := Int.emod_nonneg z.re (ne_of_gt hDi) | |
| omega⟩ | |
| let q : Fin D := ⟨(z.im % (D : ℤ)).toNat,by | |
| have h := Int.emod_lt_of_pos z.im hDi | |
| have hn := Int.emod_nonneg z.im (ne_of_gt hDi) | |
| omega⟩ | |
| refine ⟨⟨p,q⟩,?_⟩ | |
| ext <;> simp [cutResidue,finiteCutResidue,p,q,Int.emod_nonneg _ (ne_of_gt hDi)] | |
| noncomputable def vertexRepresentative (D : ℕ) | |
| (t : (Fin D × Fin D) × (Fin D × Fin D)) : ℝ × ℝ := | |
| (ordinary (finiteCutResidue D t.1)/(D : ℝ),ordinary (finiteCutResidue D t.2)/(D : ℝ)) | |
| theorem denominated_vertex_normal_form {D : ℕ} (hD : 0<D) | |
| (z : ℝ × ℝ) (u v : CutRing) | |
| (hu : ordinary u=(D : ℝ)*z.1) (hv : ordinary v=(D : ℝ)*z.2) : | |
| ∃ t : (Fin D × Fin D) × (Fin D × Fin D), ∃ x y : CutRing, | |
| z=vertexRepresentative D t+(ordinary x,ordinary y) := by | |
| obtain ⟨p,hp⟩ := cutResidue_exists_finite hD u | |
| obtain ⟨q,hq⟩ := cutResidue_exists_finite hD v | |
| have hd : (D : ℝ)≠0 := by exact_mod_cast ne_of_gt hD | |
| have hx := congrArg ordinary (cut_residue_decomposition D u) | |
| have hy := congrArg ordinary (cut_residue_decomposition D v) | |
| simp only [map_add,map_mul,map_natCast,hp,hq,hu,hv] at hx hy | |
| refine ⟨(p,q),cutQuotient D u,cutQuotient D v,?_⟩ | |
| apply Prod.ext | |
| · change z.1=ordinary (finiteCutResidue D p)/(D : ℝ)+ordinary (cutQuotient D u) | |
| apply (mul_left_cancel₀ hd) | |
| rw [mul_add,mul_div_cancel₀ _ hd] | |
| exact hx | |
| · change z.2=ordinary (finiteCutResidue D q)/(D : ℝ)+ordinary (cutQuotient D v) | |
| apply (mul_left_cancel₀ hd) | |
| rw [mul_add,mul_div_cancel₀ _ hd] | |
| exact hy | |
| theorem all_intersections_finite_templates {a : ℕ} (ha : 0<a) | |
| (i j : Fin 4) (hij : i≠j) (c d : CutRing) (z : ℝ × ℝ) | |
| (hi : cutForm a i z=ordinary c) (hj : cutForm a j z=ordinary d) : | |
| ∃ t : (Fin (commonVertexDenominator a) × Fin (commonVertexDenominator a)) × | |
| (Fin (commonVertexDenominator a) × Fin (commonVertexDenominator a)), | |
| ∃ x y : CutRing, | |
| z=vertexRepresentative (commonVertexDenominator a) t+(ordinary x,ordinary y) := by | |
| obtain ⟨u,v,hu,hv⟩ := all_intersections_common_denominator ha i j hij c d z hi hj | |
| exact denominated_vertex_normal_form (commonVertexDenominator_pos ha) z u v hu hv | |
| end VertexResidues | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |