| import OAI.Geometry.Arnold.CriticalLifts |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators ContDiff |
| open scoped Topology NNReal |
| open Set Filter |
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| namespace ArnoldCounterexample.Construction |
| open ArnoldCounterexample.Quadric ArnoldCounterexample.Field ArnoldCounterexample.Factors |
| open ArnoldCounterexample.ThreePoints |
| open scoped ContDiff Topology Matrix |
| open Set Filter |
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| lemma zeroLift_fixed : mainEquiv (project zeroLift) = project zeroLift := by |
| apply vertical_core_fixed zeroLift (by rfl) 0 (by norm_num) |
| simp only [zero_smul] |
| change core zeroVector = 0 |
| exact core_zeroVector |
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| lemma otherLift_fixed (r : ℝ) (hr : r^2 = 3) : |
| mainEquiv (project ⟨otherVector r,otherVector_unit r hr⟩) = |
| project ⟨otherVector r,otherVector_unit r hr⟩ := by |
| apply vertical_core_fixed _ (by rfl) (-(3*r/2)) |
| · have hb : |r| ≤ 2 := abs_le.mpr ⟨by nlinarith, by nlinarith⟩ |
| rw [abs_neg, abs_div, abs_mul] |
| norm_num |
| linarith |
| · exact core_otherVector r hr |
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| lemma plusLift_fixed : mainEquiv (project plusLift) = project plusLift := |
| otherLift_fixed _ (Real.sq_sqrt (by norm_num)) |
| lemma minusLift_fixed : mainEquiv (project minusLift) = project minusLift := |
| otherLift_fixed _ (by rw [neg_sq]; exact Real.sq_sqrt (by norm_num)) |
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| lemma factors_respects (z w : UnitQuadric) (h : project z = project w) : |
| sphereFactors z = sphereFactors w := by |
| obtain ⟨c,hc,he⟩ := (project_eq_iff_phase z w).mp h |
| rw [← he] |
| simp only [sphereFactors, left_scale, right_scale, hc, one_smul] |
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| lemma zeroLift_ne_plusLift : project zeroLift ≠ project plusLift := by |
| intro h |
| have he := factors_respects _ _ h |
| rw [plusLift_factors, show sphereFactors zeroLift = qzero from zeroVector_factors] at he |
| exact qzero_ne_qplus he |
| lemma zeroLift_ne_minusLift : project zeroLift ≠ project minusLift := by |
| intro h |
| have he := factors_respects _ _ h |
| rw [minusLift_factors, show sphereFactors zeroLift = qzero from zeroVector_factors] at he |
| exact qzero_ne_qminus he |
| lemma plusLift_ne_minusLift : project plusLift ≠ project minusLift := by |
| intro h |
| have he := factors_respects _ _ h |
| rw [plusLift_factors, minusLift_factors] at he |
| exact qplus_ne_qminus he |
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| lemma main_fixed_iff (q : Q3) : mainEquiv q = q ↔ |
| q = project zeroLift ∨ q = project plusLift ∨ q = project minusLift := by |
| constructor |
| · obtain ⟨z,rfl⟩ := project_surjective q |
| intro hz |
| have hz0 := fixed_zero_coordinate z hz |
| have hc := multiplierCritical_iff (sphereFactors z) |>.mp (fixed_factor_critical z hz) |
| rcases hc with hc | hc | hc |
| · exact Or.inl (sphereFactors_project_injective z zeroLift hz0 (by rfl) (hc.trans zeroVector_factors.symm)) |
| · exact Or.inr (Or.inl (sphereFactors_project_injective z plusLift hz0 (by rfl) (hc.trans plusLift_factors.symm))) |
| · exact Or.inr (Or.inr (sphereFactors_project_injective z minusLift hz0 (by rfl) (hc.trans minusLift_factors.symm))) |
| · rintro (rfl | rfl | rfl) |
| · exact zeroLift_fixed |
| · exact plusLift_fixed |
| · exact minusLift_fixed |
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| lemma main_fixed_set : fixedSet mainEquiv = {project zeroLift, project plusLift, project minusLift} := by |
| ext q |
| simp only [fixedSet, mem_ofPred_eq, main_fixed_iff, mem_insert_iff, mem_singleton_iff] |
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| lemma main_fixed_encard : (fixedSet mainEquiv).encard = 3 := by |
| rw [main_fixed_set, Set.encard_insert_of_notMem] |
| · rw [Set.encard_pair plusLift_ne_minusLift] |
| norm_num |
| · simp [zeroLift_ne_plusLift, zeroLift_ne_minusLift] |
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| end ArnoldCounterexample.Construction |
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| end |
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| end OAI |
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