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f1f7953 | 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 | 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
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