openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Sampling /CircleCharacterDynamics.lean
| import Mathlib.Topology.Instances.AddCircle.Defs | |
| import Mathlib.Basic.Real.Basic | |
| import Mathlib.Tactic | |
| namespace OAI | |
| namespace Invariant233 | |
| namespace CircleCharacterDynamics | |
| local notation "Circle" => AddCircle (1 : ℝ) | |
| local notation "rep" => (fun x : Circle => (AddCircle.equivIco (1 : ℝ) 0 x : ℝ)) | |
| theorem representative_coe (x : ℝ) : rep (x : Circle) = Int.fract x := by | |
| simp [AddCircle.coe_equivIco_mk_apply] | |
| theorem coe_representative (x : Circle) : ((rep x : ℝ) : Circle) = x := | |
| AddCircle.coe_equivIco | |
| theorem representative_add_real (x : Circle) (t : ℝ) : | |
| rep (x + (t : Circle)) = Int.fract (rep x + t) := by | |
| rw [← representative_coe] | |
| congr 1 | |
| rw [AddCircle.coe_add, coe_representative] | |
| theorem zsmul_real_phase (q : ℤ) (θ : ℝ) : | |
| q • (θ : Circle) = ((Int.fract ((q : ℝ) * θ) : ℝ) : Circle) := by | |
| rw [AddCircle.coe_fract] | |
| simpa only [zsmul_eq_mul] using | |
| (AddCircle.coe_zsmul (p := (1 : ℝ)) (n := q) (x := θ)).symm | |
| theorem character_orbit_representative (q v : ℤ) (θ : ℝ) (x : Circle) : | |
| rep (q • (x + v • (θ : Circle))) = | |
| Int.fract (rep (q • x) + (v : ℝ) * Int.fract ((q : ℝ) * θ)) := by | |
| rw [zsmul_add, smul_comm q v, zsmul_real_phase] | |
| rw [← AddCircle.coe_zsmul, zsmul_eq_mul] | |
| exact representative_add_real _ _ | |
| theorem zsmul_rational_eq_zero {q d : ℤ} (hd : d ≠ 0) (hdq : d ∣ q) (r : ℤ) : | |
| q • (((r : ℝ) / (d : ℝ) : ℝ) : Circle) = 0 := by | |
| obtain ⟨k, rfl⟩ := hdq | |
| rw [← AddCircle.coe_zsmul, zsmul_eq_mul] | |
| have hd' : (d : ℝ) ≠ 0 := by exact_mod_cast hd | |
| have he : ((d * k : ℤ) : ℝ) * ((r : ℝ) / (d : ℝ)) = ((k * r : ℤ) : ℝ) := by | |
| push_cast | |
| field_simp | |
| rw [he] | |
| apply (AddCircle.coe_eq_zero_iff (1 : ℝ)).mpr | |
| refine ⟨k * r, ?_⟩ | |
| simp [zsmul_eq_mul] | |
| theorem character_rational_translation {q d : ℤ} (hd : d ≠ 0) (hdq : d ∣ q) | |
| (r : ℤ) (x : Circle) : | |
| rep (q • (x + (((r : ℝ) / (d : ℝ) : ℝ) : Circle))) = rep (q • x) := by | |
| rw [zsmul_add, zsmul_rational_eq_zero hd hdq r, add_zero] | |
| end CircleCharacterDynamics | |
| end Invariant233 | |
| end OAI | |