File: Mathlib/Analysis/Normed/Group/AddCircle.lean

full_name: AddCircle.instNormedAddCommGroupReal
code: instance : NormedAddCommGroup (AddCircle p) 
goal: NormedAddCommGroup (AddCircle p)

full_name: AddCircle.norm_coe_mul
code: @[simp]
theorem norm_coe_mul (x : ℝ) (t : ℝ) :
    ‖(↑(t * x) : AddCircle (t * p))‖ = |t| * ‖(x : AddCircle p)‖ 
goal: ‖↑(t * x)‖ = |t| * ‖↑x‖

full_name: AddCircle.norm_neg_period
code: theorem norm_neg_period (x : ℝ) : ‖(x : AddCircle (-p))‖ = ‖(x : AddCircle p)‖ 
goal: ‖↑x‖ = ‖↑x‖

full_name: AddCircle.norm_eq_of_zero
code: @[simp]
theorem norm_eq_of_zero {x : ℝ} : ‖(x : AddCircle (0 : ℝ))‖ = |x| 
goal: ‖↑x‖ = |x|

full_name: AddCircle.norm_eq
code: theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| 
goal: ‖↑x‖ = |x - ↑(round (p⁻¹ * x)) * p|

full_name: AddCircle.norm_eq'
code: theorem norm_eq' (hp : 0 < p) {x : ℝ} : ‖(x : AddCircle p)‖ = p * |p⁻¹ * x - round (p⁻¹ * x)| 
goal: ‖↑x‖ = p * |p⁻¹ * x - ↑(round (p⁻¹ * x))|

full_name: AddCircle.norm_le_half_period
code: theorem norm_le_half_period {x : AddCircle p} (hp : p ≠ 0) : ‖x‖ ≤ |p| / 2 
goal: ‖x‖ ≤ |p| / 2

full_name: AddCircle.norm_half_period_eq
code: @[simp]
theorem norm_half_period_eq : ‖(↑(p / 2) : AddCircle p)‖ = |p| / 2 
goal: ‖↑(p / 2)‖ = |p| / 2

full_name: AddCircle.norm_coe_eq_abs_iff
code: theorem norm_coe_eq_abs_iff {x : ℝ} (hp : p ≠ 0) : ‖(x : AddCircle p)‖ = |x| ↔ |x| ≤ |p| / 2 
goal: ‖↑x‖ = |x| ↔ |x| ≤ |p| / 2

full_name: AddCircle.closedBall_eq_univ_of_half_period_le
code: theorem closedBall_eq_univ_of_half_period_le (hp : p ≠ 0) (x : AddCircle p) {ε : ℝ}
    (hε : |p| / 2 ≤ ε) : closedBall x ε = univ 
goal: Metric.closedBall x ε = Set.univ

full_name: AddCircle.coe_real_preimage_closedBall_period_zero
code: @[simp]
theorem coe_real_preimage_closedBall_period_zero (x ε : ℝ) :
    (↑) ⁻¹' closedBall (x : AddCircle (0 : ℝ)) ε = closedBall x ε 
goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε = Metric.closedBall x ε

full_name: AddCircle.coe_real_preimage_closedBall_eq_iUnion
code: theorem coe_real_preimage_closedBall_eq_iUnion (x ε : ℝ) :
    (↑) ⁻¹' closedBall (x : AddCircle p) ε = ⋃ z : ℤ, closedBall (x + z • p) ε 
goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε = ⋃ z, Metric.closedBall (x + z • p) ε

full_name: AddCircle.coe_real_preimage_closedBall_inter_eq
code: theorem coe_real_preimage_closedBall_inter_eq {x ε : ℝ} (s : Set ℝ)
    (hs : s ⊆ closedBall x (|p| / 2)) :
    (↑) ⁻¹' closedBall (x : AddCircle p) ε ∩ s = if ε < |p| / 2 then closedBall x ε ∩ s else s 
goal: QuotientAddGroup.mk ⁻¹' Metric.closedBall (↑x) ε ∩ s = if ε < |p| / 2 then Metric.closedBall x ε ∩ s else s

full_name: AddCircle.norm_div_natCast
code: theorem norm_div_natCast {m n : ℕ} :
    ‖(↑(↑m / ↑n * p) : AddCircle p)‖ = p * (↑(min (m % n) (n - m % n)) / n) 
goal: ‖↑(↑m / ↑n * p)‖ = p * (↑(min (m % n) (n - m % n)) / ↑n)

full_name: AddCircle.exists_norm_eq_of_isOfFinAddOrder
code: theorem exists_norm_eq_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u) :
    ∃ k : ℕ, ‖u‖ = p * (k / addOrderOf u) 
goal: ∃ k, ‖u‖ = p * (↑k / ↑(addOrderOf u))

full_name: AddCircle.le_add_order_smul_norm_of_isOfFinAddOrder
code: theorem le_add_order_smul_norm_of_isOfFinAddOrder {u : AddCircle p} (hu : IsOfFinAddOrder u)
    (hu' : u ≠ 0) : p ≤ addOrderOf u • ‖u‖ 
goal: p ≤ addOrderOf u • ‖u‖

full_name: UnitAddCircle.norm_eq
code: theorem norm_eq {x : ℝ} : ‖(x : UnitAddCircle)‖ = |x - round x| 
goal: ‖↑x‖ = |x - ↑(round x)|