File size: 5,111 Bytes
0dd3af4 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | 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
|