ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- toIocDiv_add_right
@[simp]
theorem toIocDiv_add_right (a b : α) : toIocDiv hp a (b + p) = toIocDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a (b + p) = toIocDiv hp a b + 1
-- toIocDiv_add_right'
@[simp]
theorem toIocDiv_add_right' (a b : α) : toIocDiv hp (a + p) b = toIocDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp (a + p) b = toIocDiv hp a b - 1
-- toIcoDiv_add_left
@[simp]
theorem toIcoDiv_add_left (a b : α) : toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a (p + b) = toIcoDiv hp a b + 1
-- toIcoDiv_add_left'
@[simp]
theorem toIcoDiv_add_left' (a b : α) : toIcoDiv hp (p + a) b = toIcoDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp (p + a) b = toIcoDiv hp a b - 1
-- toIocDiv_add_left
@[simp]
theorem toIocDiv_add_left (a b : α) : toIocDiv hp a (p + b) = toIocDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a (p + b) = toIocDiv hp a b + 1
-- toIocDiv_add_left'
@[simp]
theorem toIocDiv_add_left' (a b : α) : toIocDiv hp (p + a) b = toIocDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp (p + a) b = toIocDiv hp a b - 1
-- toIcoDiv_sub
@[simp]
theorem toIcoDiv_sub (a b : α) : toIcoDiv hp a (b - p) = toIcoDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a (b - p) = toIcoDiv hp a b - 1
-- toIcoDiv_sub'
@[simp]
theorem toIcoDiv_sub' (a b : α) : toIcoDiv hp (a - p) b = toIcoDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp (a - p) b = toIcoDiv hp a b + 1
-- toIocDiv_sub
@[simp]
theorem toIocDiv_sub (a b : α) : toIocDiv hp a (b - p) = toIocDiv hp a b - 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a (b - p) = toIocDiv hp a b - 1
-- toIocDiv_sub'
@[simp]
theorem toIocDiv_sub' (a b : α) : toIocDiv hp (a - p) b = toIocDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp (a - p) b = toIocDiv hp a b + 1
-- toIcoDiv_sub_eq_toIcoDiv_add
theorem toIcoDiv_sub_eq_toIcoDiv_add (a b c : α) :
    toIcoDiv hp a (b - c) = toIcoDiv hp (a + c) b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoDiv hp a (b - c) = toIcoDiv hp (a + c) b
-- toIocDiv_sub_eq_toIocDiv_add
theorem toIocDiv_sub_eq_toIocDiv_add (a b c : α) :
    toIocDiv hp a (b - c) = toIocDiv hp (a + c) b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIocDiv hp a (b - c) = toIocDiv hp (a + c) b
-- toIcoDiv_sub_eq_toIcoDiv_add'
theorem toIcoDiv_sub_eq_toIcoDiv_add' (a b c : α) :
    toIcoDiv hp (a - c) b = toIcoDiv hp a (b + c) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoDiv hp (a - c) b = toIcoDiv hp a (b + c)
-- toIocDiv_sub_eq_toIocDiv_add'
theorem toIocDiv_sub_eq_toIocDiv_add' (a b c : α) :
    toIocDiv hp (a - c) b = toIocDiv hp a (b + c) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIocDiv hp (a - c) b = toIocDiv hp a (b + c)
-- toIcoDiv_neg
theorem toIcoDiv_neg (a b : α) : toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a (-b) = -(toIocDiv hp (-a) b + 1)
-- toIcoDiv_neg'
theorem toIcoDiv_neg' (a b : α) : toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1)
-- toIocDiv_neg
theorem toIocDiv_neg (a b : α) : toIocDiv hp a (-b) = -(toIcoDiv hp (-a) b + 1) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a (-b) = -(toIcoDiv hp (-a) b + 1)
-- toIocDiv_neg'
theorem toIocDiv_neg' (a b : α) : toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp (-a) b = -(toIcoDiv hp a (-b) + 1)
-- toIcoMod_add_zsmul
@[simp]
theorem toIcoMod_add_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b + m • p) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp a (b + m • p) = toIcoMod hp a b
-- toIcoMod_add_zsmul'
@[simp]
theorem toIcoMod_add_zsmul' (a b : α) (m : ℤ) :
    toIcoMod hp (a + m • p) b = toIcoMod hp a b + m • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp (a + m • p) b = toIcoMod hp a b + m • p
-- toIocMod_add_zsmul
@[simp]
theorem toIocMod_add_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b + m • p) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp a (b + m • p) = toIocMod hp a b
-- toIocMod_add_zsmul'
@[simp]
theorem toIocMod_add_zsmul' (a b : α) (m : ℤ) :
    toIocMod hp (a + m • p) b = toIocMod hp a b + m • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp (a + m • p) b = toIocMod hp a b + m • p
-- toIcoMod_zsmul_add
@[simp]
theorem toIcoMod_zsmul_add (a b : α) (m : ℤ) : toIcoMod hp a (m • p + b) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp a (m • p + b) = toIcoMod hp a b
-- toIcoMod_zsmul_add'
@[simp]
theorem toIcoMod_zsmul_add' (a b : α) (m : ℤ) :
    toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp (m • p + a) b = m • p + toIcoMod hp a b
-- toIocMod_zsmul_add
@[simp]
theorem toIocMod_zsmul_add (a b : α) (m : ℤ) : toIocMod hp a (m • p + b) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp a (m • p + b) = toIocMod hp a b
-- toIocMod_zsmul_add'
@[simp]
theorem toIocMod_zsmul_add' (a b : α) (m : ℤ) :
    toIocMod hp (m • p + a) b = m • p + toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp (m • p + a) b = m • p + toIocMod hp a b
-- toIcoMod_sub_zsmul
@[simp]
theorem toIcoMod_sub_zsmul (a b : α) (m : ℤ) : toIcoMod hp a (b - m • p) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp a (b - m • p) = toIcoMod hp a b
-- toIcoMod_sub_zsmul'
@[simp]
theorem toIcoMod_sub_zsmul' (a b : α) (m : ℤ) :
    toIcoMod hp (a - m • p) b = toIcoMod hp a b - m • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIcoMod hp (a - m • p) b = toIcoMod hp a b - m • p
-- toIocMod_sub_zsmul
@[simp]
theorem toIocMod_sub_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b - m • p) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp a (b - m • p) = toIocMod hp a b
-- toIocMod_sub_zsmul'
@[simp]
theorem toIocMod_sub_zsmul' (a b : α) (m : ℤ) :
    toIocMod hp (a - m • p) b = toIocMod hp a b - m • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α) (m : ℤ)
Goal: toIocMod hp (a - m • p) b = toIocMod hp a b - m • p
-- toIcoMod_add_right
@[simp]
theorem toIcoMod_add_right (a b : α) : toIcoMod hp a (b + p) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a (b + p) = toIcoMod hp a b
-- toIcoMod_add_right'
@[simp]
theorem toIcoMod_add_right' (a b : α) : toIcoMod hp (a + p) b = toIcoMod hp a b + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp (a + p) b = toIcoMod hp a b + p
-- toIocMod_add_right
@[simp]
theorem toIocMod_add_right (a b : α) : toIocMod hp a (b + p) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a (b + p) = toIocMod hp a b
-- toIocMod_add_right'
@[simp]
theorem toIocMod_add_right' (a b : α) : toIocMod hp (a + p) b = toIocMod hp a b + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp (a + p) b = toIocMod hp a b + p
-- toIcoMod_add_left
@[simp]
theorem toIcoMod_add_left (a b : α) : toIcoMod hp a (p + b) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a (p + b) = toIcoMod hp a b
-- toIcoMod_add_left'
@[simp]
theorem toIcoMod_add_left' (a b : α) : toIcoMod hp (p + a) b = p + toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp (p + a) b = p + toIcoMod hp a b
-- toIocMod_add_left
@[simp]
theorem toIocMod_add_left (a b : α) : toIocMod hp a (p + b) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a (p + b) = toIocMod hp a b
-- toIocMod_add_left'
@[simp]
theorem toIocMod_add_left' (a b : α) : toIocMod hp (p + a) b = p + toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp (p + a) b = p + toIocMod hp a b
-- toIcoMod_sub
@[simp]
theorem toIcoMod_sub (a b : α) : toIcoMod hp a (b - p) = toIcoMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a (b - p) = toIcoMod hp a b
-- toIcoMod_sub'
@[simp]
theorem toIcoMod_sub' (a b : α) : toIcoMod hp (a - p) b = toIcoMod hp a b - p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp (a - p) b = toIcoMod hp a b - p
-- toIocMod_sub
@[simp]
theorem toIocMod_sub (a b : α) : toIocMod hp a (b - p) = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a (b - p) = toIocMod hp a b
-- toIocMod_sub'
@[simp]
theorem toIocMod_sub' (a b : α) : toIocMod hp (a - p) b = toIocMod hp a b - p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp (a - p) b = toIocMod hp a b - p
-- toIcoMod_sub_eq_sub
theorem toIcoMod_sub_eq_sub (a b c : α) : toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c
-- toIocMod_sub_eq_sub
theorem toIocMod_sub_eq_sub (a b c : α) : toIocMod hp a (b - c) = toIocMod hp (a + c) b - c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIocMod hp a (b - c) = toIocMod hp (a + c) b - c
-- toIcoMod_add_right_eq_add
theorem toIcoMod_add_right_eq_add (a b c : α) :
    toIcoMod hp a (b + c) = toIcoMod hp (a - c) b + c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoMod hp a (b + c) = toIcoMod hp (a - c) b + c
-- toIocMod_add_right_eq_add
theorem toIocMod_add_right_eq_add (a b c : α) :
    toIocMod hp a (b + c) = toIocMod hp (a - c) b + c 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIocMod hp a (b + c) = toIocMod hp (a - c) b + c
-- toIcoMod_neg
theorem toIcoMod_neg (a b : α) : toIcoMod hp a (-b) = p - toIocMod hp (-a) b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a (-b) = p - toIocMod hp (-a) b
-- toIcoMod_neg'
theorem toIcoMod_neg' (a b : α) : toIcoMod hp (-a) b = p - toIocMod hp a (-b) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp (-a) b = p - toIocMod hp a (-b)
-- toIocMod_neg
theorem toIocMod_neg (a b : α) : toIocMod hp a (-b) = p - toIcoMod hp (-a) b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a (-b) = p - toIcoMod hp (-a) b
-- toIocMod_neg'
theorem toIocMod_neg' (a b : α) : toIocMod hp (-a) b = p - toIcoMod hp a (-b) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp (-a) b = p - toIcoMod hp a (-b)
-- toIcoMod_eq_toIcoMod
theorem toIcoMod_eq_toIcoMod : toIcoMod hp a b = toIcoMod hp a c ↔ ∃ n : ℤ, c - b = n • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α}
Goal: toIcoMod hp a b = toIcoMod hp a c ↔ ∃ n, c - b = n • p
-- toIocMod_eq_toIocMod
theorem toIocMod_eq_toIocMod : toIocMod hp a b = toIocMod hp a c ↔ ∃ n : ℤ, c - b = n • p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α}
Goal: toIocMod hp a b = toIocMod hp a c ↔ ∃ n, c - b = n • p
-- AddCommGroup.modEq_iff_toIcoMod_eq_left
theorem modEq_iff_toIcoMod_eq_left : a ≡ b [PMOD p] ↔ toIcoMod hp a b = a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ toIcoMod hp a b = a
-- AddCommGroup.modEq_iff_toIocMod_eq_right
theorem modEq_iff_toIocMod_eq_right : a ≡ b [PMOD p] ↔ toIocMod hp a b = a + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ toIocMod hp a b = a + p
-- AddCommGroup.tfae_modEq
theorem tfae_modEq :
    TFAE
      [a ≡ b [PMOD p], ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
        toIcoMod hp a b + p = toIocMod hp a b] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: [a ≡ b [PMOD p], ∀ (z : ℤ), b - z • p ∉ Set.Ioo a (a + p), toIcoMod hp a b ≠ toIocMod hp a b,
    toIcoMod hp a b + p = toIocMod hp a b].TFAE
-- AddCommGroup.modEq_iff_not_forall_mem_Ioo_mod
theorem modEq_iff_not_forall_mem_Ioo_mod :
    a ≡ b [PMOD p] ↔ ∀ z : ℤ, b - z • p ∉ Set.Ioo a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ ∀ (z : ℤ), b - z • p ∉ Set.Ioo a (a + p)
-- AddCommGroup.modEq_iff_toIcoMod_ne_toIocMod
theorem modEq_iff_toIcoMod_ne_toIocMod : a ≡ b [PMOD p] ↔ toIcoMod hp a b ≠ toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ toIcoMod hp a b ≠ toIocMod hp a b
-- AddCommGroup.modEq_iff_toIcoMod_add_period_eq_toIocMod
theorem modEq_iff_toIcoMod_add_period_eq_toIocMod :
    a ≡ b [PMOD p] ↔ toIcoMod hp a b + p = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ toIcoMod hp a b + p = toIocMod hp a b
-- AddCommGroup.not_modEq_iff_toIcoMod_eq_toIocMod
theorem not_modEq_iff_toIcoMod_eq_toIocMod : ¬a ≡ b [PMOD p] ↔ toIcoMod hp a b = toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: ¬a ≡ b [PMOD p] ↔ toIcoMod hp a b = toIocMod hp a b
-- AddCommGroup.not_modEq_iff_toIcoDiv_eq_toIocDiv
theorem not_modEq_iff_toIcoDiv_eq_toIocDiv :
    ¬a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: ¬a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b
-- AddCommGroup.modEq_iff_toIcoDiv_eq_toIocDiv_add_one
theorem modEq_iff_toIcoDiv_eq_toIocDiv_add_one :
    a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: a ≡ b [PMOD p] ↔ toIcoDiv hp a b = toIocDiv hp a b + 1
-- toIcoMod_inj
@[simp]
theorem toIcoMod_inj {c : α} : toIcoMod hp c a = toIcoMod hp c b ↔ a ≡ b [PMOD p] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α}
Goal: toIcoMod hp c a = toIcoMod hp c b ↔ a ≡ b [PMOD p]
-- Ico_eq_locus_Ioc_eq_iUnion_Ioo
theorem Ico_eq_locus_Ioc_eq_iUnion_Ioo :
    { b | toIcoMod hp a b = toIocMod hp a b } = ⋃ z : ℤ, Set.Ioo (a + z • p) (a + p + z • p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a : α}
Goal: {b | toIcoMod hp a b = toIocMod hp a b} = ⋃ z, Set.Ioo (a + z • p) (a + p + z • p)
-- toIocDiv_wcovBy_toIcoDiv
theorem toIocDiv_wcovBy_toIcoDiv (a b : α) : toIocDiv hp a b ⩿ toIcoDiv hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a b ⩿ toIcoDiv hp a b
-- toIcoMod_le_toIocMod
theorem toIcoMod_le_toIocMod (a b : α) : toIcoMod hp a b ≤ toIocMod hp a b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b ≤ toIocMod hp a b
-- toIocMod_le_toIcoMod_add
theorem toIocMod_le_toIcoMod_add (a b : α) : toIocMod hp a b ≤ toIcoMod hp a b + p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b ≤ toIcoMod hp a b + p
-- toIcoMod_eq_self
theorem toIcoMod_eq_self : toIcoMod hp a b = b ↔ b ∈ Set.Ico a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: toIcoMod hp a b = b ↔ b ∈ Set.Ico a (a + p)
-- toIocMod_eq_self
theorem toIocMod_eq_self : toIocMod hp a b = b ↔ b ∈ Set.Ioc a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b : α}
Goal: toIocMod hp a b = b ↔ b ∈ Set.Ioc a (a + p)
-- toIcoMod_toIcoMod
@[simp]
theorem toIcoMod_toIcoMod (a₁ a₂ b : α) : toIcoMod hp a₁ (toIcoMod hp a₂ b) = toIcoMod hp a₁ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a₁ a₂ b : α)
Goal: toIcoMod hp a₁ (toIcoMod hp a₂ b) = toIcoMod hp a₁ b
-- toIcoMod_toIocMod
@[simp]
theorem toIcoMod_toIocMod (a₁ a₂ b : α) : toIcoMod hp a₁ (toIocMod hp a₂ b) = toIcoMod hp a₁ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a₁ a₂ b : α)
Goal: toIcoMod hp a₁ (toIocMod hp a₂ b) = toIcoMod hp a₁ b
-- toIocMod_toIocMod
@[simp]
theorem toIocMod_toIocMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIocMod hp a₂ b) = toIocMod hp a₁ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a₁ a₂ b : α)
Goal: toIocMod hp a₁ (toIocMod hp a₂ b) = toIocMod hp a₁ b
-- toIocMod_toIcoMod
@[simp]
theorem toIocMod_toIcoMod (a₁ a₂ b : α) : toIocMod hp a₁ (toIcoMod hp a₂ b) = toIocMod hp a₁ b 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a₁ a₂ b : α)
Goal: toIocMod hp a₁ (toIcoMod hp a₂ b) = toIocMod hp a₁ b
-- toIcoMod_periodic
theorem toIcoMod_periodic (a : α) : Function.Periodic (toIcoMod hp a) p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: Function.Periodic (toIcoMod hp a) p
-- toIocMod_periodic
theorem toIocMod_periodic (a : α) : Function.Periodic (toIocMod hp a) p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: Function.Periodic (toIocMod hp a) p
-- toIcoMod_zero_sub_comm
theorem toIcoMod_zero_sub_comm (a b : α) : toIcoMod hp 0 (a - b) = p - toIocMod hp 0 (b - a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp 0 (a - b) = p - toIocMod hp 0 (b - a)
-- toIocMod_zero_sub_comm
theorem toIocMod_zero_sub_comm (a b : α) : toIocMod hp 0 (a - b) = p - toIcoMod hp 0 (b - a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp 0 (a - b) = p - toIcoMod hp 0 (b - a)
-- toIcoDiv_eq_sub
theorem toIcoDiv_eq_sub (a b : α) : toIcoDiv hp a b = toIcoDiv hp 0 (b - a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a b = toIcoDiv hp 0 (b - a)
-- toIocDiv_eq_sub
theorem toIocDiv_eq_sub (a b : α) : toIocDiv hp a b = toIocDiv hp 0 (b - a) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a b = toIocDiv hp 0 (b - a)
-- toIcoMod_eq_sub
theorem toIcoMod_eq_sub (a b : α) : toIcoMod hp a b = toIcoMod hp 0 (b - a) + a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b = toIcoMod hp 0 (b - a) + a
-- toIocMod_eq_sub
theorem toIocMod_eq_sub (a b : α) : toIocMod hp a b = toIocMod hp 0 (b - a) + a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b = toIocMod hp 0 (b - a) + a
-- toIcoMod_add_toIocMod_zero
theorem toIcoMod_add_toIocMod_zero (a b : α) :
    toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - a) = p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp 0 (a - b) + toIocMod hp 0 (b - a) = p
-- toIocMod_add_toIcoMod_zero
theorem toIocMod_add_toIcoMod_zero (a b : α) :
    toIocMod hp 0 (a - b) + toIcoMod hp 0 (b - a) = p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp 0 (a - b) + toIcoMod hp 0 (b - a) = p
-- QuotientAddGroup.equivIcoMod_symm_apply
@[simps symm_apply]
def QuotientAddGroup.equivIcoMod (a : α) : α ⧸ AddSubgroup.zmultiples p ≃ Set.Ico a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α) (x : ↑(Set.Ico a (a + p)))
Goal: (QuotientAddGroup.equivIcoMod hp a).symm x = ↑↑x
-- QuotientAddGroup.equivIcoMod_coe
@[simp]
theorem QuotientAddGroup.equivIcoMod_coe (a b : α) :
    QuotientAddGroup.equivIcoMod hp a ↑b = ⟨toIcoMod hp a b, toIcoMod_mem_Ico hp a _⟩ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: (QuotientAddGroup.equivIcoMod hp a) ↑b = ⟨toIcoMod hp a b, ⋯⟩
-- QuotientAddGroup.equivIcoMod_zero
@[simp]
theorem QuotientAddGroup.equivIcoMod_zero (a : α) :
    QuotientAddGroup.equivIcoMod hp a 0 = ⟨toIcoMod hp a 0, toIcoMod_mem_Ico hp a _⟩ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: (QuotientAddGroup.equivIcoMod hp a) 0 = ⟨toIcoMod hp a 0, ⋯⟩