openAI-math-dataset / lean /OAI /GroupTheory /SimpleAmenable /Arithmetic /QuadraticRectangleCounts.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 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 | |