ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Int.ceil_lt_add_one
theorem ceil_lt_add_one (a : α) : (⌈a⌉ : α) < a + 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: ↑⌈a⌉ < a + 1
-- Int.ceil_add_le
theorem ceil_add_le (a b : α) : ⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a b : α)
Goal: ⌈a + b⌉ ≤ ⌈a⌉ + ⌈b⌉
-- Int.ceil_add_ceil_le
theorem ceil_add_ceil_le (a b : α) : ⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a b : α)
Goal: ⌈a⌉ + ⌈b⌉ ≤ ⌈a + b⌉ + 1
-- Int.ceil_pos
@[simp]
theorem ceil_pos : 0 < ⌈a⌉ ↔ 0 < a 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: 0 < ⌈a⌉ ↔ 0 < a
-- Int.ceil_zero
@[simp]
theorem ceil_zero : ⌈(0 : α)⌉ = 0 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α]
Goal: ⌈0⌉ = 0
-- Int.ceil_one
@[simp]
theorem ceil_one : ⌈(1 : α)⌉ = 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α]
Goal: ⌈1⌉ = 1
-- Int.ceil_nonneg
theorem ceil_nonneg (ha : 0 ≤ a) : 0 ≤ ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: 0 ≤ ⌈a⌉
-- Int.ceil_eq_iff
theorem ceil_eq_iff : ⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ z 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {z : ℤ} {a : α}
Goal: ⌈a⌉ = z ↔ ↑z - 1 < a ∧ a ≤ ↑z
-- Int.ceil_eq_zero_iff
@[simp]
theorem ceil_eq_zero_iff : ⌈a⌉ = 0 ↔ a ∈ Ioc (-1 : α) 0 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: ⌈a⌉ = 0 ↔ a ∈ Set.Ioc (-1) 0
-- Int.ceil_eq_on_Ioc
theorem ceil_eq_on_Ioc (z : ℤ) : ∀ a ∈ Set.Ioc (z - 1 : α) z, ⌈a⌉ = z 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (z : ℤ) (a : α)
Goal: a ∈ Set.Ioc (↑z - 1) ↑z → ⌈a⌉ = z
-- Int.ceil_eq_on_Ioc'
theorem ceil_eq_on_Ioc' (z : ℤ) : ∀ a ∈ Set.Ioc (z - 1 : α) z, (⌈a⌉ : α) = z 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (z : ℤ) (a : α)
Goal: a ∈ Set.Ioc (↑z - 1) ↑z → ↑⌈a⌉ = ↑z
-- Int.floor_le_ceil
theorem floor_le_ceil (a : α) : ⌊a⌋ ≤ ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: ⌊a⌋ ≤ ⌈a⌉
-- Int.floor_lt_ceil_of_lt
theorem floor_lt_ceil_of_lt {a b : α} (h : a < b) : ⌊a⌋ < ⌈b⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a b : α} (h : a < b)
Goal: ⌊a⌋ < ⌈b⌉
-- Int.preimage_ceil_singleton
@[simp]
theorem preimage_ceil_singleton (m : ℤ) : (ceil : α → ℤ) ⁻¹' {m} = Ioc ((m : α) - 1) m 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (m : ℤ)
Goal: Int.ceil ⁻¹' {m} = Set.Ioc (↑m - 1) ↑m
-- Int.fract_eq_zero_or_add_one_sub_ceil
theorem fract_eq_zero_or_add_one_sub_ceil (a : α) : fract a = 0 ∨ fract a = a + 1 - (⌈a⌉ : α) 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: Int.fract a = 0 ∨ Int.fract a = a + 1 - ↑⌈a⌉
-- Int.ceil_eq_add_one_sub_fract
theorem ceil_eq_add_one_sub_fract (ha : fract a ≠ 0) : (⌈a⌉ : α) = a + 1 - fract a 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : Int.fract a ≠ 0)
Goal: ↑⌈a⌉ = a + 1 - Int.fract a
-- Int.ceil_sub_self_eq
theorem ceil_sub_self_eq (ha : fract a ≠ 0) : (⌈a⌉ : α) - a = 1 - fract a 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : Int.fract a ≠ 0)
Goal: ↑⌈a⌉ - a = 1 - Int.fract a
-- Int.preimage_Ioo
@[simp]
theorem preimage_Ioo {a b : α} : ((↑) : ℤ → α) ⁻¹' Set.Ioo a b = Set.Ioo ⌊a⌋ ⌈b⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a b : α}
Goal: Int.cast ⁻¹' Set.Ioo a b = Set.Ioo ⌊a⌋ ⌈b⌉
-- Int.preimage_Ico
@[simp]
theorem preimage_Ico {a b : α} : ((↑) : ℤ → α) ⁻¹' Set.Ico a b = Set.Ico ⌈a⌉ ⌈b⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a b : α}
Goal: Int.cast ⁻¹' Set.Ico a b = Set.Ico ⌈a⌉ ⌈b⌉
-- Int.preimage_Ioc
@[simp]
theorem preimage_Ioc {a b : α} : ((↑) : ℤ → α) ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋ ⌊b⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a b : α}
Goal: Int.cast ⁻¹' Set.Ioc a b = Set.Ioc ⌊a⌋ ⌊b⌋
-- Int.preimage_Icc
@[simp]
theorem preimage_Icc {a b : α} : ((↑) : ℤ → α) ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉ ⌊b⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a b : α}
Goal: Int.cast ⁻¹' Set.Icc a b = Set.Icc ⌈a⌉ ⌊b⌋
-- Int.preimage_Ioi
@[simp]
theorem preimage_Ioi : ((↑) : ℤ → α) ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: Int.cast ⁻¹' Set.Ioi a = Set.Ioi ⌊a⌋
-- Int.preimage_Ici
@[simp]
theorem preimage_Ici : ((↑) : ℤ → α) ⁻¹' Set.Ici a = Set.Ici ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: Int.cast ⁻¹' Set.Ici a = Set.Ici ⌈a⌉
-- Int.preimage_Iio
@[simp]
theorem preimage_Iio : ((↑) : ℤ → α) ⁻¹' Set.Iio a = Set.Iio ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: Int.cast ⁻¹' Set.Iio a = Set.Iio ⌈a⌉
-- Int.preimage_Iic
@[simp]
theorem preimage_Iic : ((↑) : ℤ → α) ⁻¹' Set.Iic a = Set.Iic ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: Int.cast ⁻¹' Set.Iic a = Set.Iic ⌊a⌋
-- round_zero
@[simp]
theorem round_zero : round (0 : α) = 0 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α]
Goal: round 0 = 0
-- round_one
@[simp]
theorem round_one : round (1 : α) = 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α]
Goal: round 1 = 1
-- round_natCast
@[simp]
theorem round_natCast (n : ℕ) : round (n : α) = n 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (n : ℕ)
Goal: round ↑n = ↑n
-- round_ofNat
@[simp]
theorem round_ofNat (n : ℕ) [n.AtLeastTwo] : round (no_index (OfNat.ofNat n : α)) = n 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (n : ℕ) [n.AtLeastTwo]
Goal: round (OfNat.ofNat n) = ↑n
-- round_intCast
@[simp]
theorem round_intCast (n : ℤ) : round (n : α) = n 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (n : ℤ)
Goal: round ↑n = n
-- round_add_int
@[simp]
theorem round_add_int (x : α) (y : ℤ) : round (x + y) = round x + y 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℤ)
Goal: round (x + ↑y) = round x + y
-- round_add_one
@[simp]
theorem round_add_one (a : α) : round (a + 1) = round a + 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: round (a + 1) = round a + 1
-- round_sub_int
@[simp]
theorem round_sub_int (x : α) (y : ℤ) : round (x - y) = round x - y 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℤ)
Goal: round (x - ↑y) = round x - y
-- round_sub_one
@[simp]
theorem round_sub_one (a : α) : round (a - 1) = round a - 1 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: round (a - 1) = round a - 1
-- round_add_nat
@[simp]
theorem round_add_nat (x : α) (y : ℕ) : round (x + y) = round x + y 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℕ)
Goal: round (x + ↑y) = round x + ↑y
-- round_add_ofNat
@[simp]
theorem round_add_ofNat (x : α) (n : ℕ) [n.AtLeastTwo] :
    round (x + (no_index (OfNat.ofNat n))) = round x + OfNat.ofNat n 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (n : ℕ) [n.AtLeastTwo]
Goal: round (x + OfNat.ofNat n) = round x + OfNat.ofNat n
-- round_sub_nat
@[simp]
theorem round_sub_nat (x : α) (y : ℕ) : round (x - y) = round x - y 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℕ)
Goal: round (x - ↑y) = round x - ↑y
-- round_sub_ofNat
@[simp]
theorem round_sub_ofNat (x : α) (n : ℕ) [n.AtLeastTwo] :
    round (x - (no_index (OfNat.ofNat n))) = round x - OfNat.ofNat n 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (n : ℕ) [n.AtLeastTwo]
Goal: round (x - OfNat.ofNat n) = round x - OfNat.ofNat n
-- round_int_add
@[simp]
theorem round_int_add (x : α) (y : ℤ) : round ((y : α) + x) = y + round x 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℤ)
Goal: round (↑y + x) = y + round x
-- round_nat_add
@[simp]
theorem round_nat_add (x : α) (y : ℕ) : round ((y : α) + x) = y + round x 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (y : ℕ)
Goal: round (↑y + x) = ↑y + round x
-- round_ofNat_add
@[simp]
theorem round_ofNat_add (n : ℕ) [n.AtLeastTwo] (x : α) :
    round ((no_index (OfNat.ofNat n)) + x) = OfNat.ofNat n + round x 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (n : ℕ) [n.AtLeastTwo] (x : α)
Goal: round (OfNat.ofNat n + x) = OfNat.ofNat n + round x
-- abs_sub_round_eq_min
theorem abs_sub_round_eq_min (x : α) : |x - round x| = min (fract x) (1 - fract x) 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α)
Goal: |x - ↑(round x)| = min (Int.fract x) (1 - Int.fract x)
-- round_le
theorem round_le (x : α) (z : ℤ) : |x - round x| ≤ |x - z| 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (x : α) (z : ℤ)
Goal: |x - ↑(round x)| ≤ |x - ↑z|
-- round_eq
theorem round_eq (x : α) : round x = ⌊x + 1 / 2⌋ 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] (x : α)
Goal: round x = ⌊x + 1 / 2⌋
-- round_two_inv
@[simp]
theorem round_two_inv : round (2⁻¹ : α) = 1 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α]
Goal: round 2⁻¹ = 1
-- round_neg_two_inv
@[simp]
theorem round_neg_two_inv : round (-2⁻¹ : α) = 0 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α]
Goal: round (-2⁻¹) = 0
-- round_eq_zero_iff
@[simp]
theorem round_eq_zero_iff {x : α} : round x = 0 ↔ x ∈ Ico (-(1 / 2)) ((1 : α) / 2) 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] {x : α}
Goal: round x = 0 ↔ x ∈ Set.Ico (-(1 / 2)) (1 / 2)
-- abs_sub_round
theorem abs_sub_round (x : α) : |x - round x| ≤ 1 / 2 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] (x : α)
Goal: |x - ↑(round x)| ≤ 1 / 2
-- abs_sub_round_div_natCast_eq
theorem abs_sub_round_div_natCast_eq {m n : ℕ} :
    |(m : α) / n - round ((m : α) / n)| = ↑(min (m % n) (n - m % n)) / n 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] {m n : ℕ}
Goal: |↑m / ↑n - ↑(round (↑m / ↑n))| = ↑(min (m % n) (n - m % n)) / ↑n
-- sub_half_lt_round
theorem sub_half_lt_round (x : α) : x - 1 / 2 < round x 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] (x : α)
Goal: x - 1 / 2 < ↑(round x)
-- round_le_add_half
theorem round_le_add_half (x : α) : round x ≤ x + 1 / 2 
Arguments: {α : Type u_2} [LinearOrderedField α] [FloorRing α] (x : α)
Goal: ↑(round x) ≤ x + 1 / 2
-- Nat.floor_congr
theorem floor_congr (h : ∀ n : ℕ, (n : α) ≤ a ↔ (n : β) ≤ b) : ⌊a⌋₊ = ⌊b⌋₊ 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] [FloorSemiring α] [FloorSemiring β] {a : α} {b : β} (h : ∀ (n : ℕ), ↑n ≤ a ↔ ↑n ≤ b)
Goal: ⌊a⌋₊ = ⌊b⌋₊
-- Nat.ceil_congr
theorem ceil_congr (h : ∀ n : ℕ, a ≤ n ↔ b ≤ n) : ⌈a⌉₊ = ⌈b⌉₊ 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] [FloorSemiring α] [FloorSemiring β] {a : α} {b : β} (h : ∀ (n : ℕ), a ≤ ↑n ↔ b ≤ ↑n)
Goal: ⌈a⌉₊ = ⌈b⌉₊
-- Nat.map_floor
theorem map_floor (f : F) (hf : StrictMono f) (a : α) : ⌊f a⌋₊ = ⌊a⌋₊ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] [FloorSemiring α] [FloorSemiring β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: ⌊f a⌋₊ = ⌊a⌋₊
-- Nat.map_ceil
theorem map_ceil (f : F) (hf : StrictMono f) (a : α) : ⌈f a⌉₊ = ⌈a⌉₊ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedSemiring α] [LinearOrderedSemiring β] [FloorSemiring α] [FloorSemiring β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: ⌈f a⌉₊ = ⌈a⌉₊
-- Int.floor_congr
theorem floor_congr (h : ∀ n : ℤ, (n : α) ≤ a ↔ (n : β) ≤ b) : ⌊a⌋ = ⌊b⌋ 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedRing β] [FloorRing α] [FloorRing β] {a : α} {b : β} (h : ∀ (n : ℤ), ↑n ≤ a ↔ ↑n ≤ b)
Goal: ⌊a⌋ = ⌊b⌋
-- Int.ceil_congr
theorem ceil_congr (h : ∀ n : ℤ, a ≤ n ↔ b ≤ n) : ⌈a⌉ = ⌈b⌉ 
Arguments: {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedRing β] [FloorRing α] [FloorRing β] {a : α} {b : β} (h : ∀ (n : ℤ), a ≤ ↑n ↔ b ≤ ↑n)
Goal: ⌈a⌉ = ⌈b⌉
-- Int.map_floor
theorem map_floor (f : F) (hf : StrictMono f) (a : α) : ⌊f a⌋ = ⌊a⌋ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedRing β] [FloorRing α] [FloorRing β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: ⌊f a⌋ = ⌊a⌋
-- Int.map_ceil
theorem map_ceil (f : F) (hf : StrictMono f) (a : α) : ⌈f a⌉ = ⌈a⌉ 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedRing β] [FloorRing α] [FloorRing β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: ⌈f a⌉ = ⌈a⌉
-- Int.map_fract
theorem map_fract (f : F) (hf : StrictMono f) (a : α) : fract (f a) = f (fract a) 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedRing α] [LinearOrderedRing β] [FloorRing α] [FloorRing β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: Int.fract (f a) = f (Int.fract a)
-- Int.map_round
theorem map_round (f : F) (hf : StrictMono f) (a : α) : round (f a) = round a 
Arguments: {F : Type u_1} {α : Type u_2} {β : Type u_3} [LinearOrderedField α] [LinearOrderedField β] [FloorRing α] [FloorRing β] [FunLike F α β] [RingHomClass F α β] (f : F) (hf : StrictMono ⇑f) (a : α)
Goal: round (f a) = round a
-- Int.floor_toNat
theorem Int.floor_toNat (a : α) : ⌊a⌋.toNat = ⌊a⌋₊ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: ⌊a⌋.toNat = ⌊a⌋₊
-- Int.ceil_toNat
theorem Int.ceil_toNat (a : α) : ⌈a⌉.toNat = ⌈a⌉₊ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] (a : α)
Goal: ⌈a⌉.toNat = ⌈a⌉₊
-- Nat.floor_int
@[simp]
theorem Nat.floor_int : (Nat.floor : ℤ → ℕ) = Int.toNat 
Goal: Nat.floor = Int.toNat
-- Nat.ceil_int
@[simp]
theorem Nat.ceil_int : (Nat.ceil : ℤ → ℕ) = Int.toNat 
Goal: Nat.ceil = Int.toNat
-- Int.ofNat_floor_eq_floor
theorem Int.ofNat_floor_eq_floor (ha : 0 ≤ a) : (⌊a⌋₊ : ℤ) = ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: ↑⌊a⌋₊ = ⌊a⌋
-- Int.ofNat_ceil_eq_ceil
theorem Int.ofNat_ceil_eq_ceil (ha : 0 ≤ a) : (⌈a⌉₊ : ℤ) = ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: ↑⌈a⌉₊ = ⌈a⌉
-- natCast_floor_eq_intCast_floor
theorem natCast_floor_eq_intCast_floor (ha : 0 ≤ a) : (⌊a⌋₊ : α) = ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: ↑⌊a⌋₊ = ↑⌊a⌋
-- natCast_ceil_eq_intCast_ceil
theorem natCast_ceil_eq_intCast_ceil  (ha : 0 ≤ a) : (⌈a⌉₊ : α) = ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: ↑⌈a⌉₊ = ↑⌈a⌉
-- subsingleton_floorRing
theorem subsingleton_floorRing {α} [LinearOrderedRing α] : Subsingleton (FloorRing α) 
Arguments: {α : Type u_4} [LinearOrderedRing α]
Goal: Subsingleton (FloorRing α)
-- _private.Mathlib.Algebra.Order.Floor.0.Mathlib.Meta.Positivity.int_floor_nonneg
private theorem int_floor_nonneg [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a) :
    0 ≤ ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 ≤ a)
Goal: 0 ≤ ⌊a⌋
-- _private.Mathlib.Algebra.Order.Floor.0.Mathlib.Meta.Positivity.int_floor_nonneg_of_pos
private theorem int_floor_nonneg_of_pos [LinearOrderedRing α] [FloorRing α] {a : α}
    (ha : 0 < a) :
    0 ≤ ⌊a⌋ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α} (ha : 0 < a)
Goal: 0 ≤ ⌊a⌋
-- _private.Mathlib.Algebra.Order.Floor.0.Mathlib.Meta.Positivity.nat_ceil_pos
private theorem nat_ceil_pos [LinearOrderedSemiring α] [FloorSemiring α] {a : α} :
    0 < a → 0 < ⌈a⌉₊ 
Arguments: {α : Type u_2} [LinearOrderedSemiring α] [FloorSemiring α] {a : α}
Goal: 0 < a → 0 < ⌈a⌉₊
-- _private.Mathlib.Algebra.Order.Floor.0.Mathlib.Meta.Positivity.int_ceil_pos
private theorem int_ceil_pos [LinearOrderedRing α] [FloorRing α] {a : α} : 0 < a → 0 < ⌈a⌉ 
Arguments: {α : Type u_2} [LinearOrderedRing α] [FloorRing α] {a : α}
Goal: 0 < a → 0 < ⌈a⌉
-- mabs_le'
@[to_additive] lemma mabs_le' : |a|ₘ ≤ b ↔ a ≤ b ∧ a⁻¹ ≤ b 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a b : α}
Goal: mabs a ≤ b ↔ a ≤ b ∧ a⁻¹ ≤ b
-- abs_le'
@[to_additive] lemma mabs_le' : |a|ₘ ≤ b ↔ a ≤ b ∧ a⁻¹ ≤ b 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a b : α}
Goal: |a| ≤ b ↔ a ≤ b ∧ -a ≤ b
-- le_abs_self
@[to_additive] lemma le_mabs_self (a : α) : a ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: a ≤ |a|
-- le_mabs_self
@[to_additive] lemma le_mabs_self (a : α) : a ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a ≤ mabs a
-- neg_le_abs
@[to_additive] lemma inv_le_mabs (a : α) : a⁻¹ ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: -a ≤ |a|
-- inv_le_mabs
@[to_additive] lemma inv_le_mabs (a : α) : a⁻¹ ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: a⁻¹ ≤ mabs a
-- mabs_le_mabs
@[to_additive] lemma mabs_le_mabs (h₀ : a ≤ b) (h₁ : a⁻¹ ≤ b) : |a|ₘ ≤ |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a b : α} (h₀ : a ≤ b) (h₁ : a⁻¹ ≤ b)
Goal: mabs a ≤ mabs b
-- abs_le_abs
@[to_additive] lemma mabs_le_mabs (h₀ : a ≤ b) (h₁ : a⁻¹ ≤ b) : |a|ₘ ≤ |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a b : α} (h₀ : a ≤ b) (h₁ : -a ≤ b)
Goal: |a| ≤ |b|
-- mabs_inv
@[to_additive (attr := simp)] lemma mabs_inv (a : α) : |a⁻¹|ₘ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a : α)
Goal: mabs a⁻¹ = mabs a
-- abs_neg
@[to_additive (attr := simp)] lemma mabs_inv (a : α) : |a⁻¹|ₘ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a : α)
Goal: |(-a)| = |a|
-- abs_sub_comm
@[to_additive] lemma mabs_div_comm (a b : α) : |a / b|ₘ = |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] (a b : α)
Goal: |a - b| = |b - a|
-- mabs_div_comm
@[to_additive] lemma mabs_div_comm (a b : α) : |a / b|ₘ = |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] (a b : α)
Goal: mabs (a / b) = mabs (b / a)
-- mabs_of_one_le
@[to_additive] lemma mabs_of_one_le (h : 1 ≤ a) : |a|ₘ = a 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : 1 ≤ a)
Goal: mabs a = a
-- abs_of_nonneg
@[to_additive] lemma mabs_of_one_le (h : 1 ≤ a) : |a|ₘ = a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : 0 ≤ a)
Goal: |a| = a
-- abs_of_pos
@[to_additive] lemma mabs_of_one_lt (h : 1 < a) : |a|ₘ = a 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : 0 < a)
Goal: |a| = a
-- mabs_of_one_lt
@[to_additive] lemma mabs_of_one_lt (h : 1 < a) : |a|ₘ = a 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : 1 < a)
Goal: mabs a = a
-- mabs_of_le_one
@[to_additive] lemma mabs_of_le_one (h : a ≤ 1) : |a|ₘ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : a ≤ 1)
Goal: mabs a = a⁻¹
-- abs_of_nonpos
@[to_additive] lemma mabs_of_le_one (h : a ≤ 1) : |a|ₘ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a ≤ 0)
Goal: |a| = -a
-- mabs_of_lt_one
@[to_additive] lemma mabs_of_lt_one (h : a < 1) : |a|ₘ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (h : a < 1)
Goal: mabs a = a⁻¹
-- abs_of_neg
@[to_additive] lemma mabs_of_lt_one (h : a < 1) : |a|ₘ = a⁻¹ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (h : a < 0)
Goal: |a| = -a
-- abs_le_abs_of_nonneg
@[to_additive] lemma mabs_le_mabs_of_one_le (ha : 1 ≤ a) (hab : a ≤ b) : |a|ₘ ≤ |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] {a b : α} [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (ha : 0 ≤ a) (hab : a ≤ b)
Goal: |a| ≤ |b|
-- mabs_le_mabs_of_one_le
@[to_additive] lemma mabs_le_mabs_of_one_le (ha : 1 ≤ a) (hab : a ≤ b) : |a|ₘ ≤ |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] {a b : α} [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (ha : 1 ≤ a) (hab : a ≤ b)
Goal: mabs a ≤ mabs b
-- mabs_one
@[to_additive (attr := simp)] lemma mabs_one : |(1 : α)|ₘ = 1 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1]
Goal: mabs 1 = 1
-- abs_zero
@[to_additive (attr := simp)] lemma mabs_one : |(1 : α)|ₘ = 1 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1]
Goal: |0| = 0
-- abs_nonneg
@[to_additive (attr := simp) abs_nonneg] lemma one_le_mabs (a : α) : 1 ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: 0 ≤ |a|
-- one_le_mabs
@[to_additive (attr := simp) abs_nonneg] lemma one_le_mabs (a : α) : 1 ≤ |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: 1 ≤ mabs a
-- abs_abs
@[to_additive (attr := simp)] lemma mabs_mabs (a : α) : |(|a|ₘ)|ₘ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: |(|a|)| = |a|
-- mabs_mabs
@[to_additive (attr := simp)] lemma mabs_mabs (a : α) : |(|a|ₘ)|ₘ = |a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [Group α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] [CovariantClass α α (Function.swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a : α)
Goal: mabs (mabs a) = mabs a
-- abs_add_le
@[to_additive "The absolute value satisfies the triangle inequality."]
lemma mabs_mul_le (a b : α) : |a * b|ₘ ≤ |a|ₘ * |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: |a + b| ≤ |a| + |b|
-- mabs_mul_le
@[to_additive "The absolute value satisfies the triangle inequality."]
lemma mabs_mul_le (a b : α) : |a * b|ₘ ≤ |a|ₘ * |b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: mabs (a * b) ≤ mabs a * mabs b
-- abs_abs_sub_abs_le
@[to_additive]
lemma mabs_mabs_div_mabs_le (a b : α) : |(|a|ₘ / |b|ₘ)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: ||a| - |b|| ≤ |a - b|
-- mabs_mabs_div_mabs_le
@[to_additive]
lemma mabs_mabs_div_mabs_le (a b : α) : |(|a|ₘ / |b|ₘ)|ₘ ≤ |a / b|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: mabs (mabs a / mabs b) ≤ mabs (a / b)
-- sup_sub_inf_eq_abs_sub
@[to_additive] lemma sup_div_inf_eq_mabs_div (a b : α) : (a ⊔ b) / (a ⊓ b) = |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a ⊔ b - a ⊓ b = |b - a|
-- sup_div_inf_eq_mabs_div
@[to_additive] lemma sup_div_inf_eq_mabs_div (a b : α) : (a ⊔ b) / (a ⊓ b) = |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: (a ⊔ b) / (a ⊓ b) = mabs (b / a)
-- two_nsmul_sup_eq_add_add_abs_sub
@[to_additive two_nsmul_sup_eq_add_add_abs_sub]
lemma sup_sq_eq_mul_mul_mabs_div (a b : α) : (a ⊔ b) ^ 2 = a * b * |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [AddCommGroup α] [CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: 2 • (a ⊔ b) = a + b + |b - a|
-- sup_sq_eq_mul_mul_mabs_div
@[to_additive two_nsmul_sup_eq_add_add_abs_sub]
lemma sup_sq_eq_mul_mul_mabs_div (a b : α) : (a ⊔ b) ^ 2 = a * b * |b / a|ₘ 
Arguments: {α : Type u_1} [Lattice α] [CommGroup α] [CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: (a ⊔ b) ^ 2 = a * b * mabs (b / a)