openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /PolygonGeometry /FlagSeparatingLevels.lean
| import Mathlib | |
| import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.FlagGerms | |
| import OAI.GroupTheory.SimpleAmenable.Arithmetic.QuadraticRectangleCounts | |
| namespace OAI | |
| section | |
| section | |
| open scoped symmDiff | |
| namespace SimpleAmenable | |
| open scoped commutatorElement | |
| open scoped commutatorElement | |
| section FlagSeparatingLevels | |
| open Classical | |
| def FlagSeparates {a : ℕ} {v : ℝ×ℝ} (j : Fin 4) (c : CutRing) | |
| (z w : SquareFlag a v) : Prop := | |
| ¬ (flagMem ⟨halfPlane a j c,halfPlane_mem a j c⟩ z ↔ | |
| flagMem ⟨halfPlane a j c,halfPlane_mem a j c⟩ w) | |
| theorem flagSeparates_ordinary_bound {a : ℕ} {v : ℝ×ℝ} {j : Fin 4} {c : CutRing} | |
| {z w : SquareFlag a v} (h : FlagSeparates j c z w) : | |
| ordinary c ∈ Set.Icc (min (cutForm a j z.val) (cutForm a j w.val)) | |
| (max (cutForm a j z.val) (cutForm a j w.val)) := by | |
| unfold FlagSeparates at h | |
| simp only [flagMem_halfPlane] at h | |
| constructor | |
| · by_contra hn | |
| have hz : ordinary c < cutForm a j z.val := lt_of_lt_of_le (lt_of_not_ge hn) (min_le_left _ _) | |
| have hw : ordinary c < cutForm a j w.val := lt_of_lt_of_le (lt_of_not_ge hn) (min_le_right _ _) | |
| simp [ne_of_gt hz,ne_of_gt hw,not_lt_of_ge hz.le,not_lt_of_ge hw.le] at h | |
| · by_contra hn | |
| have hz : cutForm a j z.val < ordinary c := lt_of_le_of_lt (le_max_left _ _) (lt_of_not_ge hn) | |
| have hw : cutForm a j w.val < ordinary c := lt_of_le_of_lt (le_max_right _ _) (lt_of_not_ge hn) | |
| simp [ne_of_lt hz,ne_of_lt hw,hz,hw] at h | |
| theorem flagSeparates_bounded_finite {a : ℕ} {v : ℝ×ℝ} (j : Fin 4) | |
| (z w : SquareFlag a v) (R : ℝ) : | |
| {c : CutRing | FlagSeparates j c z w ∧ |conjugate c| ≤ R}.Finite := by | |
| let b := |R|+1 | |
| have hb : 0 < b := by dsimp [b]; positivity | |
| let x := min (cutForm a j z.val) (cutForm a j w.val) | |
| let y := max (cutForm a j z.val) (cutForm a j w.val) | |
| apply (cutRectangle_finite x (-b) (y-x) (2*b) (by positivity)).subset | |
| intro c hc | |
| have ho := flagSeparates_ordinary_bound hc.1 | |
| have ht := abs_le.mp hc.2 | |
| change _ ∧ _ | |
| constructor | |
| · simpa [x,y] using ho | |
| · constructor <;> dsimp [b] <;> linarith [le_abs_self R] | |
| theorem flagSeparates_exists_min {a : ℕ} {v : ℝ×ℝ} (j : Fin 4) | |
| (z w : SquareFlag a v) (h : ∃ c,FlagSeparates j c z w) : | |
| ∃ c,FlagSeparates j c z w ∧ ∀ d,FlagSeparates j d z w → | |
| |conjugate c| ≤ |conjugate d| := by | |
| obtain ⟨c,hc⟩ := h | |
| let S : Set CutRing := {d : CutRing | FlagSeparates j d z w ∧ |conjugate d| ≤ |conjugate c|} | |
| have hS : S.Finite := flagSeparates_bounded_finite j z w _ | |
| have hn : S.Nonempty := ⟨c,hc,le_rfl⟩ | |
| obtain ⟨d,hd,hmin⟩ := Set.exists_min_image S (fun e => |conjugate e|) hS hn | |
| refine ⟨d,hd.1,fun e he => ?_⟩ | |
| by_cases hec : |conjugate e| ≤ |conjugate c| | |
| · exact hmin e ⟨he,hec⟩ | |
| · exact hd.2.trans (le_of_not_ge hec) | |
| theorem flagMem_halfPlane_local_translation {a : ℕ} {v : ℝ×ℝ} | |
| (u : CutRing×CutRing) (j : Fin 4) (c : CutRing) (z z' : SquareFlag a v) | |
| (hz : z'.val=z.val+(ordinary u.1,ordinary u.2)) : | |
| flagMem ⟨halfPlane a j (c+integralCutForm a j u),halfPlane_mem _ _ _⟩ z' ↔ | |
| flagMem ⟨halfPlane a j c,halfPlane_mem _ _ _⟩ z := by | |
| rw [flagMem_halfPlane,flagMem_halfPlane,hz,cutForm_add,cutForm_ordinary,map_add] | |
| simp only [add_left_inj,add_lt_add_iff_right] | |
| theorem flagSeparates_local_translation {a : ℕ} {v : ℝ×ℝ} | |
| (u : CutRing×CutRing) (j : Fin 4) (c : CutRing) (z w z' w' : SquareFlag a v) | |
| (hz : z'.val=z.val+(ordinary u.1,ordinary u.2)) | |
| (hw : w'.val=w.val+(ordinary u.1,ordinary u.2)) : | |
| FlagSeparates j (c+integralCutForm a j u) z' w' ↔ FlagSeparates j c z w := by | |
| unfold FlagSeparates | |
| rw [flagMem_halfPlane_local_translation u j c z z' hz, | |
| flagMem_halfPlane_local_translation u j c w w' hw] | |
| theorem separating_min_translation_bound {a : ℕ} {v : ℝ×ℝ} | |
| (u : CutRing×CutRing) (j : Fin 4) (z w z' w' : SquareFlag a v) | |
| (hz : z'.val=z.val+(ordinary u.1,ordinary u.2)) | |
| (hw : w'.val=w.val+(ordinary u.1,ordinary u.2)) | |
| (c d : CutRing) (hc : FlagSeparates j c z w) (hd : FlagSeparates j d z' w') | |
| (hcmin : ∀ e,FlagSeparates j e z w → |conjugate c| ≤ |conjugate e|) | |
| (hdmin : ∀ e,FlagSeparates j e z' w' → |conjugate d| ≤ |conjugate e|) : | |
| abs (|conjugate c|-|conjugate d|) ≤ |conjugate (integralCutForm a j u)| := by | |
| have hd' := hdmin (c+integralCutForm a j u) | |
| ((flagSeparates_local_translation u j c z w z' w' hz hw).mpr hc) | |
| have hc' := hcmin (d-integralCutForm a j u) (by | |
| apply (flagSeparates_local_translation u j (d-integralCutForm a j u) z w z' w' hz hw).mp | |
| simpa only [sub_add_cancel] using hd) | |
| have h₁ := abs_add_le (conjugate c) (conjugate (integralCutForm a j u)) | |
| have h₂ : |conjugate d-conjugate (integralCutForm a j u)| ≤ |conjugate d|+|conjugate (integralCutForm a j u)| := by | |
| simpa only [sub_zero,zero_sub,abs_neg] using abs_sub_le (conjugate d) 0 (conjugate (integralCutForm a j u)) | |
| simp only [map_add,map_sub] at hd' hc' | |
| apply abs_le.mpr | |
| constructor <;> linarith | |
| end FlagSeparatingLevels | |
| end SimpleAmenable | |
| end | |
| end | |
| end OAI | |