File: Mathlib/Algebra/Order/Interval/Set/Group.lean

full_name: Set.inv_mem_Icc_iff
code: @[to_additive]
theorem inv_mem_Icc_iff : a⁻¹ ∈ Set.Icc c d ↔ a ∈ Set.Icc d⁻¹ c⁻¹ 
goal: a⁻¹ ∈ Set.Icc c d ↔ a ∈ Set.Icc d⁻¹ c⁻¹

full_name: Set.neg_mem_Icc_iff
code: @[to_additive]
theorem inv_mem_Icc_iff : a⁻¹ ∈ Set.Icc c d ↔ a ∈ Set.Icc d⁻¹ c⁻¹ 
goal: -a ∈ Set.Icc c d ↔ a ∈ Set.Icc (-d) (-c)

full_name: Set.neg_mem_Ico_iff
code: @[to_additive]
theorem inv_mem_Ico_iff : a⁻¹ ∈ Set.Ico c d ↔ a ∈ Set.Ioc d⁻¹ c⁻¹ 
goal: -a ∈ Set.Ico c d ↔ a ∈ Set.Ioc (-d) (-c)

full_name: Set.inv_mem_Ico_iff
code: @[to_additive]
theorem inv_mem_Ico_iff : a⁻¹ ∈ Set.Ico c d ↔ a ∈ Set.Ioc d⁻¹ c⁻¹ 
goal: a⁻¹ ∈ Set.Ico c d ↔ a ∈ Set.Ioc d⁻¹ c⁻¹

full_name: Set.inv_mem_Ioc_iff
code: @[to_additive]
theorem inv_mem_Ioc_iff : a⁻¹ ∈ Set.Ioc c d ↔ a ∈ Set.Ico d⁻¹ c⁻¹ 
goal: a⁻¹ ∈ Set.Ioc c d ↔ a ∈ Set.Ico d⁻¹ c⁻¹

full_name: Set.neg_mem_Ioc_iff
code: @[to_additive]
theorem inv_mem_Ioc_iff : a⁻¹ ∈ Set.Ioc c d ↔ a ∈ Set.Ico d⁻¹ c⁻¹ 
goal: -a ∈ Set.Ioc c d ↔ a ∈ Set.Ico (-d) (-c)

full_name: Set.neg_mem_Ioo_iff
code: @[to_additive]
theorem inv_mem_Ioo_iff : a⁻¹ ∈ Set.Ioo c d ↔ a ∈ Set.Ioo d⁻¹ c⁻¹ 
goal: -a ∈ Set.Ioo c d ↔ a ∈ Set.Ioo (-d) (-c)

full_name: Set.inv_mem_Ioo_iff
code: @[to_additive]
theorem inv_mem_Ioo_iff : a⁻¹ ∈ Set.Ioo c d ↔ a ∈ Set.Ioo d⁻¹ c⁻¹ 
goal: a⁻¹ ∈ Set.Ioo c d ↔ a ∈ Set.Ioo d⁻¹ c⁻¹

full_name: Set.add_mem_Icc_iff_left
code: theorem add_mem_Icc_iff_left : a + b ∈ Set.Icc c d ↔ a ∈ Set.Icc (c - b) (d - b) 
goal: a + b ∈ Set.Icc c d ↔ a ∈ Set.Icc (c - b) (d - b)

full_name: Set.add_mem_Ico_iff_left
code: theorem add_mem_Ico_iff_left : a + b ∈ Set.Ico c d ↔ a ∈ Set.Ico (c - b) (d - b) 
goal: a + b ∈ Set.Ico c d ↔ a ∈ Set.Ico (c - b) (d - b)

full_name: Set.add_mem_Ioc_iff_left
code: theorem add_mem_Ioc_iff_left : a + b ∈ Set.Ioc c d ↔ a ∈ Set.Ioc (c - b) (d - b) 
goal: a + b ∈ Set.Ioc c d ↔ a ∈ Set.Ioc (c - b) (d - b)

full_name: Set.add_mem_Ioo_iff_left
code: theorem add_mem_Ioo_iff_left : a + b ∈ Set.Ioo c d ↔ a ∈ Set.Ioo (c - b) (d - b) 
goal: a + b ∈ Set.Ioo c d ↔ a ∈ Set.Ioo (c - b) (d - b)

full_name: Set.add_mem_Icc_iff_right
code: theorem add_mem_Icc_iff_right : a + b ∈ Set.Icc c d ↔ b ∈ Set.Icc (c - a) (d - a) 
goal: a + b ∈ Set.Icc c d ↔ b ∈ Set.Icc (c - a) (d - a)

full_name: Set.add_mem_Ico_iff_right
code: theorem add_mem_Ico_iff_right : a + b ∈ Set.Ico c d ↔ b ∈ Set.Ico (c - a) (d - a) 
goal: a + b ∈ Set.Ico c d ↔ b ∈ Set.Ico (c - a) (d - a)

full_name: Set.add_mem_Ioc_iff_right
code: theorem add_mem_Ioc_iff_right : a + b ∈ Set.Ioc c d ↔ b ∈ Set.Ioc (c - a) (d - a) 
goal: a + b ∈ Set.Ioc c d ↔ b ∈ Set.Ioc (c - a) (d - a)

full_name: Set.add_mem_Ioo_iff_right
code: theorem add_mem_Ioo_iff_right : a + b ∈ Set.Ioo c d ↔ b ∈ Set.Ioo (c - a) (d - a) 
goal: a + b ∈ Set.Ioo c d ↔ b ∈ Set.Ioo (c - a) (d - a)

full_name: Set.sub_mem_Icc_iff_left
code: theorem sub_mem_Icc_iff_left : a - b ∈ Set.Icc c d ↔ a ∈ Set.Icc (c + b) (d + b) 
goal: a - b ∈ Set.Icc c d ↔ a ∈ Set.Icc (c + b) (d + b)

full_name: Set.sub_mem_Ico_iff_left
code: theorem sub_mem_Ico_iff_left : a - b ∈ Set.Ico c d ↔ a ∈ Set.Ico (c + b) (d + b) 
goal: a - b ∈ Set.Ico c d ↔ a ∈ Set.Ico (c + b) (d + b)

full_name: Set.sub_mem_Ioc_iff_left
code: theorem sub_mem_Ioc_iff_left : a - b ∈ Set.Ioc c d ↔ a ∈ Set.Ioc (c + b) (d + b) 
goal: a - b ∈ Set.Ioc c d ↔ a ∈ Set.Ioc (c + b) (d + b)

full_name: Set.sub_mem_Ioo_iff_left
code: theorem sub_mem_Ioo_iff_left : a - b ∈ Set.Ioo c d ↔ a ∈ Set.Ioo (c + b) (d + b) 
goal: a - b ∈ Set.Ioo c d ↔ a ∈ Set.Ioo (c + b) (d + b)

full_name: Set.sub_mem_Icc_iff_right
code: theorem sub_mem_Icc_iff_right : a - b ∈ Set.Icc c d ↔ b ∈ Set.Icc (a - d) (a - c) 
goal: a - b ∈ Set.Icc c d ↔ b ∈ Set.Icc (a - d) (a - c)

full_name: Set.sub_mem_Ico_iff_right
code: theorem sub_mem_Ico_iff_right : a - b ∈ Set.Ico c d ↔ b ∈ Set.Ioc (a - d) (a - c) 
goal: a - b ∈ Set.Ico c d ↔ b ∈ Set.Ioc (a - d) (a - c)

full_name: Set.sub_mem_Ioc_iff_right
code: theorem sub_mem_Ioc_iff_right : a - b ∈ Set.Ioc c d ↔ b ∈ Set.Ico (a - d) (a - c) 
goal: a - b ∈ Set.Ioc c d ↔ b ∈ Set.Ico (a - d) (a - c)

full_name: Set.sub_mem_Ioo_iff_right
code: theorem sub_mem_Ioo_iff_right : a - b ∈ Set.Ioo c d ↔ b ∈ Set.Ioo (a - d) (a - c) 
goal: a - b ∈ Set.Ioo c d ↔ b ∈ Set.Ioo (a - d) (a - c)

full_name: Set.mem_Icc_iff_abs_le
code: theorem mem_Icc_iff_abs_le {R : Type*} [LinearOrderedAddCommGroup R] {x y z : R} :
    |x - y| ≤ z ↔ y ∈ Icc (x - z) (x + z) 
goal: |x - y| ≤ z ↔ y ∈ Set.Icc (x - z) (x + z)

full_name: Set.nonempty_Ico_sdiff
code: /-- If we remove a smaller interval from a larger, the result is nonempty -/
theorem nonempty_Ico_sdiff {x dx y dy : α} (h : dy < dx) (hx : 0 < dx) :
    Nonempty ↑(Ico x (x + dx) \ Ico y (y + dy)) 
goal: Nonempty ↑(Set.Ico x (x + dx) \ Set.Ico y (y + dy))

full_name: Set.pairwise_disjoint_Ioc_mul_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ioc_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioc (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (a * b ^ n) (a * b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ioc_add_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ioc_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioc (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (a + n • b) (a + (n + 1) • b))

full_name: Set.pairwise_disjoint_Ico_add_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ico_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ico (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (a + n • b) (a + (n + 1) • b))

full_name: Set.pairwise_disjoint_Ico_mul_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ico_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ico (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (a * b ^ n) (a * b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ioo_add_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ioo_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioo (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (a + n • b) (a + (n + 1) • b))

full_name: Set.pairwise_disjoint_Ioo_mul_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ioo_mul_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioo (a * b ^ n) (a * b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (a * b ^ n) (a * b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ioc_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ioc_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioc (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (b ^ n) (b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ioc_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ioc_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioc (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (n • b) ((n + 1) • b))

full_name: Set.pairwise_disjoint_Ico_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ico_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ico (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (b ^ n) (b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ico_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ico_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ico (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ico (n • b) ((n + 1) • b))

full_name: Set.pairwise_disjoint_Ioo_zpow
code: @[to_additive]
theorem pairwise_disjoint_Ioo_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioo (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (b ^ n) (b ^ (n + 1)))

full_name: Set.pairwise_disjoint_Ioo_zsmul
code: @[to_additive]
theorem pairwise_disjoint_Ioo_zpow :
    Pairwise (Disjoint on fun n : ℤ => Ioo (b ^ n) (b ^ (n + 1))) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (n • b) ((n + 1) • b))

full_name: Set.pairwise_disjoint_Ioc_add_intCast
code: theorem pairwise_disjoint_Ioc_add_intCast :
    Pairwise (Disjoint on fun n : ℤ => Ioc (a + n) (a + n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (a + ↑n) (a + ↑n + 1))

full_name: Set.pairwise_disjoint_Ico_add_intCast
code: theorem pairwise_disjoint_Ico_add_intCast :
    Pairwise (Disjoint on fun n : ℤ => Ico (a + n) (a + n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ico (a + ↑n) (a + ↑n + 1))

full_name: Set.pairwise_disjoint_Ioo_add_intCast
code: theorem pairwise_disjoint_Ioo_add_intCast :
    Pairwise (Disjoint on fun n : ℤ => Ioo (a + n) (a + n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (a + ↑n) (a + ↑n + 1))

full_name: Set.pairwise_disjoint_Ico_intCast
code: theorem pairwise_disjoint_Ico_intCast :
    Pairwise (Disjoint on fun n : ℤ => Ico (n : α) (n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ico (↑n) (↑n + 1))

full_name: Set.pairwise_disjoint_Ioo_intCast
code: theorem pairwise_disjoint_Ioo_intCast : Pairwise (Disjoint on fun n : ℤ => Ioo (n : α) (n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ioo (↑n) (↑n + 1))

full_name: Set.pairwise_disjoint_Ioc_intCast
code: theorem pairwise_disjoint_Ioc_intCast : Pairwise (Disjoint on fun n : ℤ => Ioc (n : α) (n + 1)) 
goal: Pairwise (Disjoint on fun n => Set.Ioc (↑n) (↑n + 1))