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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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.Arithmetic.PolygonArithmetic
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section QuadraticRectangleCounts
open Classical
theorem conjugateStrip_floor_inj (y b : ℝ) (hb : 0<b) :
Set.InjOn (fun u : CutRing => ⌊b*ordinary u⌋)
{u | conjugate u∈Set.Icc y (y+b)} := by
intro u hu w hw he
change ⌊b*ordinary u⌋=⌊b*ordinary w⌋ at he
by_contra hne
have hn := one_le_abs_cut_norm (sub_ne_zero.mpr hne)
simp only [map_sub,abs_mul] at hn
have hc : |conjugate u-conjugate w|≤b := by
apply abs_le.mpr
constructor <;> linarith [hu.1,hu.2,hw.1,hw.2]
have h₁ := Int.floor_le (b*ordinary u)
have h₂ := Int.lt_floor_add_one (b*ordinary u)
have h₃ := Int.floor_le (b*ordinary w)
have h₄ := Int.lt_floor_add_one (b*ordinary w)
rw [he] at h₁ h₂
have hlt : |b*(ordinary u-ordinary w)|<1 := by
apply abs_lt.mpr
constructor <;> nlinarith
rw [abs_mul,abs_of_pos hb] at hlt
nlinarith [abs_nonneg (ordinary u-ordinary w)]
def cutRectangle (x y a b : ℝ) : Set CutRing :=
{u | ordinary u∈Set.Icc x (x+a) ∧ conjugate u∈Set.Icc y (y+b)}
theorem cutRectangle_finite (x y a b : ℝ) (hb : 0<b) : (cutRectangle x y a b).Finite := by
have him : ((fun u : CutRing => ⌊b*ordinary u⌋) '' cutRectangle x y a b).Finite := by
apply (Set.finite_Icc ⌊b*x⌋ ⌊b*(x+a)⌋).subset
rintro _ ⟨u,hu,rfl⟩
exact ⟨Int.floor_mono (mul_le_mul_of_nonneg_left hu.1.1 hb.le),
Int.floor_mono (mul_le_mul_of_nonneg_left hu.1.2 hb.le)⟩
exact him.of_finite_image (fun _ hu _ hw he =>
conjugateStrip_floor_inj y b hb hu.2 hw.2 he)
theorem cutRectangle_card_floor (S : Finset CutRing) (x y a b : ℝ) (hb : 0<b)
(hS : ∀u∈S,u∈cutRectangle x y a b) :
S.card≤(⌊b*(x+a)⌋+1-⌊b*x⌋).toNat := by
have hinj : Set.InjOn (fun u : CutRing => ⌊b*ordinary u⌋) (↑S) := by
intro u hu w hw he
exact conjugateStrip_floor_inj y b hb (hS u hu).2 (hS w hw).2 he
rw [← Int.card_Icc,← Finset.card_image_of_injOn hinj]
apply Finset.card_le_card
intro t ht
obtain ⟨u,hu,rfl⟩ := Finset.mem_image.mp ht
exact Finset.mem_Icc.mpr
⟨Int.floor_mono (mul_le_mul_of_nonneg_left (hS u hu).1.1 hb.le),
Int.floor_mono (mul_le_mul_of_nonneg_left (hS u hu).1.2 hb.le)⟩
theorem cutRectangle_card_bound (S : Finset CutRing) (x y a b : ℝ)
(ha : 0≤a) (hb : 0<b) (hS : ∀u∈S,u∈cutRectangle x y a b) :
(S.card:ℝ)<a*b+2 := by
have hc := cutRectangle_card_floor S x y a b hb hS
have hle : ⌊b*x⌋≤⌊b*(x+a)⌋ := Int.floor_mono (by nlinarith)
have hnon : 0≤⌊b*(x+a)⌋+1-⌊b*x⌋ := by omega
have hc' : (S.card:ℤ)≤⌊b*(x+a)⌋+1-⌊b*x⌋ := by
rw [← Int.toNat_of_nonneg hnon]
exact_mod_cast hc
have hc'' : (S.card:ℝ)≤(⌊b*(x+a)⌋:ℝ)+1-(⌊b*x⌋:ℝ) := by exact_mod_cast hc'
have h₁ := Int.floor_le (b*(x+a))
have h₂ := Int.lt_floor_add_one (b*x)
nlinarith
end QuadraticRectangleCounts
end SimpleAmenable
end
end
end OAI
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