ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- QuotientAddGroup.equivIocMod_symm_apply
@[simps symm_apply]
def QuotientAddGroup.equivIocMod (a : α) : α ⧸ AddSubgroup.zmultiples p ≃ Set.Ioc a (a + p) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α) (x : ↑(Set.Ioc a (a + p)))
Goal: (QuotientAddGroup.equivIocMod hp a).symm x = ↑↑x
-- QuotientAddGroup.equivIocMod_coe
@[simp]
theorem QuotientAddGroup.equivIocMod_coe (a b : α) :
    QuotientAddGroup.equivIocMod hp a ↑b = ⟨toIocMod hp a b, toIocMod_mem_Ioc hp a _⟩ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b : α)
Goal: (QuotientAddGroup.equivIocMod hp a) ↑b = ⟨toIocMod hp a b, ⋯⟩
-- QuotientAddGroup.equivIocMod_zero
@[simp]
theorem QuotientAddGroup.equivIocMod_zero (a : α) :
    QuotientAddGroup.equivIocMod hp a 0 = ⟨toIocMod hp a 0, toIocMod_mem_Ioc hp a _⟩ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: (QuotientAddGroup.equivIocMod hp a) 0 = ⟨toIocMod hp a 0, ⋯⟩
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_iff
private theorem toIxxMod_iff (x₁ x₂ x₃ : α) : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ↔
    toIcoMod hp 0 (x₂ - x₁) + toIcoMod hp 0 (x₁ - x₃) ≤ p 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (x₁ x₂ x₃ : α)
Goal: toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ↔ toIcoMod hp 0 (x₂ - x₁) + toIcoMod hp 0 (x₁ - x₃) ≤ p
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_cyclic_left
private theorem toIxxMod_cyclic_left {x₁ x₂ x₃ : α} (h : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃) :
    toIcoMod hp x₂ x₃ ≤ toIocMod hp x₂ x₁ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {x₁ x₂ x₃ : α} (h : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃)
Goal: toIcoMod hp x₂ x₃ ≤ toIocMod hp x₂ x₁
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_antisymm
private theorem toIxxMod_antisymm (h₁₂₃ : toIcoMod hp a b ≤ toIocMod hp a c)
    (h₁₃₂ : toIcoMod hp a c ≤ toIocMod hp a b) :
    b ≡ a [PMOD p] ∨ c ≡ b [PMOD p] ∨ a ≡ c [PMOD p] 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {a b c : α} (h₁₂₃ : toIcoMod hp a b ≤ toIocMod hp a c) (h₁₃₂ : toIcoMod hp a c ≤ toIocMod hp a b)
Goal: b ≡ a [PMOD p] ∨ c ≡ b [PMOD p] ∨ a ≡ c [PMOD p]
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_total'
private theorem toIxxMod_total' (a b c : α) :
    toIcoMod hp b a ≤ toIocMod hp b c ∨ toIcoMod hp b c ≤ toIocMod hp b a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoMod hp b a ≤ toIocMod hp b c ∨ toIcoMod hp b c ≤ toIocMod hp b a
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_total
private theorem toIxxMod_total (a b c : α) :
    toIcoMod hp a b ≤ toIocMod hp a c ∨ toIcoMod hp c b ≤ toIocMod hp c a 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) (a b c : α)
Goal: toIcoMod hp a b ≤ toIocMod hp a c ∨ toIcoMod hp c b ≤ toIocMod hp c a
-- _private.Mathlib.Algebra.Order.ToIntervalMod.0.toIxxMod_trans
private theorem toIxxMod_trans {x₁ x₂ x₃ x₄ : α}
    (h₁₂₃ : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ∧ ¬toIcoMod hp x₃ x₂ ≤ toIocMod hp x₃ x₁)
    (h₂₃₄ : toIcoMod hp x₂ x₄ ≤ toIocMod hp x₂ x₃ ∧ ¬toIcoMod hp x₃ x₄ ≤ toIocMod hp x₃ x₂) :
    toIcoMod hp x₁ x₄ ≤ toIocMod hp x₁ x₃ ∧ ¬toIcoMod hp x₃ x₄ ≤ toIocMod hp x₃ x₁ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} (hp : 0 < p) {x₁ x₂ x₃ x₄ : α} (h₁₂₃ : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ∧ ¬toIcoMod hp x₃ x₂ ≤ toIocMod hp x₃ x₁) (h₂₃₄ : toIcoMod hp x₂ x₄ ≤ toIocMod hp x₂ x₃ ∧ ¬toIcoMod hp x₃ x₄ ≤ toIocMod hp x₃ x₂)
Goal: toIcoMod hp x₁ x₄ ≤ toIocMod hp x₁ x₃ ∧ ¬toIcoMod hp x₃ x₄ ≤ toIocMod hp x₃ x₁
-- QuotientAddGroup.btw_coe_iff'
theorem btw_coe_iff' {x₁ x₂ x₃ : α} :
    Btw.btw (x₁ : α ⧸ AddSubgroup.zmultiples p) x₂ x₃ ↔
      toIcoMod hp'.out 0 (x₂ - x₁) ≤ toIocMod hp'.out 0 (x₃ - x₁) 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} [hp' : Fact (0 < p)] {x₁ x₂ x₃ : α}
Goal: btw ↑x₁ ↑x₂ ↑x₃ ↔ toIcoMod ⋯ 0 (x₂ - x₁) ≤ toIocMod ⋯ 0 (x₃ - x₁)
-- QuotientAddGroup.btw_coe_iff
theorem btw_coe_iff {x₁ x₂ x₃ : α} :
    Btw.btw (x₁ : α ⧸ AddSubgroup.zmultiples p) x₂ x₃ ↔
      toIcoMod hp'.out x₁ x₂ ≤ toIocMod hp'.out x₁ x₃ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [hα : Archimedean α] {p : α} [hp' : Fact (0 < p)] {x₁ x₂ x₃ : α}
Goal: btw ↑x₁ ↑x₂ ↑x₃ ↔ toIcoMod ⋯ x₁ x₂ ≤ toIocMod ⋯ x₁ x₃
-- toIcoDiv_eq_floor
theorem toIcoDiv_eq_floor (a b : α) : toIcoDiv hp a b = ⌊(b - a) / p⌋ 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoDiv hp a b = ⌊(b - a) / p⌋
-- toIocDiv_eq_neg_floor
theorem toIocDiv_eq_neg_floor (a b : α) : toIocDiv hp a b = -⌊(a + p - b) / p⌋ 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocDiv hp a b = -⌊(a + p - b) / p⌋
-- toIcoDiv_zero_one
theorem toIcoDiv_zero_one (b : α) : toIcoDiv (zero_lt_one' α) 0 b = ⌊b⌋ 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] (b : α)
Goal: toIcoDiv ⋯ 0 b = ⌊b⌋
-- toIcoMod_eq_add_fract_mul
theorem toIcoMod_eq_add_fract_mul (a b : α) :
    toIcoMod hp a b = a + Int.fract ((b - a) / p) * p 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIcoMod hp a b = a + Int.fract ((b - a) / p) * p
-- toIcoMod_eq_fract_mul
theorem toIcoMod_eq_fract_mul (b : α) : toIcoMod hp 0 b = Int.fract (b / p) * p 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] {p : α} (hp : 0 < p) (b : α)
Goal: toIcoMod hp 0 b = Int.fract (b / p) * p
-- toIocMod_eq_sub_fract_mul
theorem toIocMod_eq_sub_fract_mul (a b : α) :
    toIocMod hp a b = a + p - Int.fract ((a + p - b) / p) * p 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] {p : α} (hp : 0 < p) (a b : α)
Goal: toIocMod hp a b = a + p - Int.fract ((a + p - b) / p) * p
-- toIcoMod_zero_one
theorem toIcoMod_zero_one (b : α) : toIcoMod (zero_lt_one' α) 0 b = Int.fract b 
Arguments: {α : Type u_1} [LinearOrderedField α] [FloorRing α] (b : α)
Goal: toIcoMod ⋯ 0 b = Int.fract b
-- iUnion_Ioc_add_zsmul
theorem iUnion_Ioc_add_zsmul : ⋃ n : ℤ, Ioc (a + n • p) (a + (n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: ⋃ n, Set.Ioc (a + n • p) (a + (n + 1) • p) = Set.univ
-- iUnion_Ico_add_zsmul
theorem iUnion_Ico_add_zsmul : ⋃ n : ℤ, Ico (a + n • p) (a + (n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: ⋃ n, Set.Ico (a + n • p) (a + (n + 1) • p) = Set.univ
-- iUnion_Icc_add_zsmul
theorem iUnion_Icc_add_zsmul : ⋃ n : ℤ, Icc (a + n • p) (a + (n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p) (a : α)
Goal: ⋃ n, Set.Icc (a + n • p) (a + (n + 1) • p) = Set.univ
-- iUnion_Ioc_zsmul
theorem iUnion_Ioc_zsmul : ⋃ n : ℤ, Ioc (n • p) ((n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p)
Goal: ⋃ n, Set.Ioc (n • p) ((n + 1) • p) = Set.univ
-- iUnion_Ico_zsmul
theorem iUnion_Ico_zsmul : ⋃ n : ℤ, Ico (n • p) ((n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p)
Goal: ⋃ n, Set.Ico (n • p) ((n + 1) • p) = Set.univ
-- iUnion_Icc_zsmul
theorem iUnion_Icc_zsmul : ⋃ n : ℤ, Icc (n • p) ((n + 1) • p) = univ 
Arguments: {α : Type u_1} [LinearOrderedAddCommGroup α] [Archimedean α] {p : α} (hp : 0 < p)
Goal: ⋃ n, Set.Icc (n • p) ((n + 1) • p) = Set.univ
-- iUnion_Ioc_add_intCast
theorem iUnion_Ioc_add_intCast : ⋃ n : ℤ, Ioc (a + n) (a + n + 1) = Set.univ 
Arguments: {α : Type u_1} [LinearOrderedRing α] [Archimedean α] (a : α)
Goal: ⋃ n, Set.Ioc (a + ↑n) (a + ↑n + 1) = Set.univ
-- iUnion_Ico_add_intCast
theorem iUnion_Ico_add_intCast : ⋃ n : ℤ, Ico (a + n) (a + n + 1) = Set.univ 
Arguments: {α : Type u_1} [LinearOrderedRing α] [Archimedean α] (a : α)
Goal: ⋃ n, Set.Ico (a + ↑n) (a + ↑n + 1) = Set.univ
-- iUnion_Icc_add_intCast
theorem iUnion_Icc_add_intCast : ⋃ n : ℤ, Icc (a + n) (a + n + 1) = Set.univ 
Arguments: {α : Type u_1} [LinearOrderedRing α] [Archimedean α] (a : α)
Goal: ⋃ n, Set.Icc (a + ↑n) (a + ↑n + 1) = Set.univ
-- iUnion_Ioc_intCast
theorem iUnion_Ioc_intCast : ⋃ n : ℤ, Ioc (n : α) (n + 1) = Set.univ 
Arguments: (α : Type u_1) [LinearOrderedRing α] [Archimedean α]
Goal: ⋃ n, Set.Ioc (↑n) (↑n + 1) = Set.univ
-- iUnion_Ico_intCast
theorem iUnion_Ico_intCast : ⋃ n : ℤ, Ico (n : α) (n + 1) = Set.univ 
Arguments: (α : Type u_1) [LinearOrderedRing α] [Archimedean α]
Goal: ⋃ n, Set.Ico (↑n) (↑n + 1) = Set.univ
-- iUnion_Icc_intCast
theorem iUnion_Icc_intCast : ⋃ n : ℤ, Icc (n : α) (n + 1) = Set.univ 
Arguments: (α : Type u_1) [LinearOrderedRing α] [Archimedean α]
Goal: ⋃ n, Set.Icc (↑n) (↑n + 1) = Set.univ
-- _private.Mathlib.Algebra.Order.Ring.Star.0.StarOrderedRing.mul_le_mul_of_nonneg_left
private lemma mul_le_mul_of_nonneg_left {R : Type*} [CommSemiring R] [PartialOrder R]
    [StarRing R] [StarOrderedRing R] {a b c : R} (hab : a ≤ b) (hc : 0 ≤ c) : c * a ≤ c * b 
Arguments: {R : Type u_1} [CommSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {a b c : R} (hab : a ≤ b) (hc : 0 ≤ c)
Goal: c * a ≤ c * b
-- OrderedSemiring.zero_le_one
class OrderedSemiring (α : Type u) extends Semiring α, OrderedAddCommMonoid α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_le_mul_of_nonneg_left : ∀ a b c : α, a ≤ b → 0 ≤ c → c * a ≤ c * b
    protected mul_le_mul_of_nonneg_right : ∀ a b c : α, a ≤ b → 0 ≤ c → a * c ≤ b * c
Arguments: {α : Type u} [self : OrderedSemiring α]
Goal: 0 ≤ 1
-- OrderedSemiring.mul_le_mul_of_nonneg_left
class OrderedSemiring (α : Type u) extends Semiring α, OrderedAddCommMonoid α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_le_mul_of_nonneg_left : ∀ a b c : α, a ≤ b → 0 ≤ c → c * a ≤ c * b
    protected mul_le_mul_of_nonneg_right : ∀ a b c : α, a ≤ b → 0 ≤ c → a * c ≤ b * c
Arguments: {α : Type u} [self : OrderedSemiring α] (a b c : α)
Goal: a ≤ b → 0 ≤ c → c * a ≤ c * b
-- OrderedSemiring.mul_le_mul_of_nonneg_right
class OrderedSemiring (α : Type u) extends Semiring α, OrderedAddCommMonoid α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_le_mul_of_nonneg_left : ∀ a b c : α, a ≤ b → 0 ≤ c → c * a ≤ c * b
    protected mul_le_mul_of_nonneg_right : ∀ a b c : α, a ≤ b → 0 ≤ c → a * c ≤ b * c
Arguments: {α : Type u} [self : OrderedSemiring α] (a b c : α)
Goal: a ≤ b → 0 ≤ c → a * c ≤ b * c
-- OrderedRing.zero_le_one
class OrderedRing (α : Type u) extends Ring α, OrderedAddCommGroup α where
    protected zero_le_one : 0 ≤ (1 : α)
    protected mul_nonneg : ∀ a b : α, 0 ≤ a → 0 ≤ b → 0 ≤ a * b
Arguments: {α : Type u} [self : OrderedRing α]
Goal: 0 ≤ 1
-- OrderedRing.mul_nonneg
class OrderedRing (α : Type u) extends Ring α, OrderedAddCommGroup α where
    protected zero_le_one : 0 ≤ (1 : α)
    protected mul_nonneg : ∀ a b : α, 0 ≤ a → 0 ≤ b → 0 ≤ a * b
Arguments: {α : Type u} [self : OrderedRing α] (a b : α)
Goal: 0 ≤ a → 0 ≤ b → 0 ≤ a * b
-- StrictOrderedSemiring.zero_le_one
class StrictOrderedSemiring (α : Type u) extends Semiring α, OrderedCancelAddCommMonoid α,
    Nontrivial α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b
    protected mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c
Arguments: {α : Type u} [self : StrictOrderedSemiring α]
Goal: 0 ≤ 1
-- StrictOrderedSemiring.mul_lt_mul_of_pos_right
class StrictOrderedSemiring (α : Type u) extends Semiring α, OrderedCancelAddCommMonoid α,
    Nontrivial α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b
    protected mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c
Arguments: {α : Type u} [self : StrictOrderedSemiring α] (a b c : α)
Goal: a < b → 0 < c → a * c < b * c
-- StrictOrderedSemiring.mul_lt_mul_of_pos_left
class StrictOrderedSemiring (α : Type u) extends Semiring α, OrderedCancelAddCommMonoid α,
    Nontrivial α where
    protected zero_le_one : (0 : α) ≤ 1
    protected mul_lt_mul_of_pos_left : ∀ a b c : α, a < b → 0 < c → c * a < c * b
    protected mul_lt_mul_of_pos_right : ∀ a b c : α, a < b → 0 < c → a * c < b * c
Arguments: {α : Type u} [self : StrictOrderedSemiring α] (a b c : α)
Goal: a < b → 0 < c → c * a < c * b
-- StrictOrderedRing.mul_pos
class StrictOrderedRing (α : Type u) extends Ring α, OrderedAddCommGroup α, Nontrivial α where
    protected zero_le_one : 0 ≤ (1 : α)
    protected mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b
Arguments: {α : Type u} [self : StrictOrderedRing α] (a b : α)
Goal: 0 < a → 0 < b → 0 < a * b
-- StrictOrderedRing.zero_le_one
class StrictOrderedRing (α : Type u) extends Ring α, OrderedAddCommGroup α, Nontrivial α where
    protected zero_le_one : 0 ≤ (1 : α)
    protected mul_pos : ∀ a b : α, 0 < a → 0 < b → 0 < a * b
Arguments: {α : Type u} [self : StrictOrderedRing α]
Goal: 0 ≤ 1
-- CanonicallyOrderedCommSemiring.eq_zero_or_eq_zero_of_mul_eq_zero
class CanonicallyOrderedCommSemiring (α : Type*) extends CanonicallyOrderedAddCommMonoid α,
    CommSemiring α where
    protected eq_zero_or_eq_zero_of_mul_eq_zero : ∀ {a b : α}, a * b = 0 → a = 0 ∨ b = 0
Arguments: {α : Type u_2} [self : CanonicallyOrderedCommSemiring α] {a b : α}
Goal: a * b = 0 → a = 0 ∨ b = 0
-- Odd.pos
lemma Odd.pos [Nontrivial α] : Odd a → 0 < a 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} [Nontrivial α]
Goal: Odd a → 0 < a
-- CanonicallyOrderedCommSemiring.mul_pos
@[simp]
protected theorem mul_pos : 0 < a * b ↔ 0 < a ∧ 0 < b 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a b : α}
Goal: 0 < a * b ↔ 0 < a ∧ 0 < b
-- CanonicallyOrderedCommSemiring.pow_pos
lemma pow_pos (ha : 0 < a) (n : ℕ) : 0 < a ^ n 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a : α} (ha : 0 < a) (n : ℕ)
Goal: 0 < a ^ n
-- CanonicallyOrderedCommSemiring.mul_lt_mul_of_lt_of_lt
protected lemma mul_lt_mul_of_lt_of_lt [PosMulStrictMono α] (hab : a < b) (hcd : c < d) :
    a * c < b * d 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a b c d : α} [PosMulStrictMono α] (hab : a < b) (hcd : c < d)
Goal: a * c < b * d
-- AddLECancellable.mul_tsub
protected theorem mul_tsub (h : AddLECancellable (a * c)) : a * (b - c) = a * b - a * c 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a b c : α} [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] (h : AddLECancellable (a * c))
Goal: a * (b - c) = a * b - a * c
-- AddLECancellable.tsub_mul
protected theorem tsub_mul (h : AddLECancellable (b * c)) : (a - b) * c = a * c - b * c 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] {a b c : α} [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] (h : AddLECancellable (b * c))
Goal: (a - b) * c = a * c - b * c
-- mul_tsub
theorem mul_tsub (a b c : α) : a * (b - c) = a * b - a * c 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: a * (b - c) = a * b - a * c
-- tsub_mul
theorem tsub_mul (a b c : α) : (a - b) * c = a * c - b * c 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b c : α)
Goal: (a - b) * c = a * c - b * c
-- mul_tsub_one
lemma mul_tsub_one (a b : α) : a * (b - 1) = a * b - a 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: a * (b - 1) = a * b - a
-- tsub_one_mul
lemma tsub_one_mul (a b : α) : (a - 1) * b = a * b - b 
Arguments: {α : Type u} [CanonicallyOrderedCommSemiring α] [Sub α] [OrderedSub α] [IsTotal α fun x x_1 => x ≤ x_1] [ContravariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1] (a b : α)
Goal: (a - 1) * b = a * b - b
-- Finset.map_add_left_Icc
@[simp] lemma map_add_left_Icc (a b c : α) :
    (Icc a b).map (addLeftEmbedding c) = Icc (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addLeftEmbedding c) (Finset.Icc a b) = Finset.Icc (c + a) (c + b)
-- Finset.map_add_right_Icc
@[simp] lemma map_add_right_Icc (a b c : α) :
    (Icc a b).map (addRightEmbedding c) = Icc (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addRightEmbedding c) (Finset.Icc a b) = Finset.Icc (a + c) (b + c)
-- Finset.map_add_left_Ico
@[simp] lemma map_add_left_Ico (a b c : α) :
    (Ico a b).map (addLeftEmbedding c) = Ico (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addLeftEmbedding c) (Finset.Ico a b) = Finset.Ico (c + a) (c + b)
-- Finset.map_add_right_Ico
@[simp] lemma map_add_right_Ico (a b c : α) :
    (Ico a b).map (addRightEmbedding c) = Ico (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addRightEmbedding c) (Finset.Ico a b) = Finset.Ico (a + c) (b + c)
-- Finset.map_add_left_Ioc
@[simp] lemma map_add_left_Ioc (a b c : α) :
    (Ioc a b).map (addLeftEmbedding c) = Ioc (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addLeftEmbedding c) (Finset.Ioc a b) = Finset.Ioc (c + a) (c + b)
-- Finset.map_add_right_Ioc
@[simp] lemma map_add_right_Ioc (a b c : α) :
    (Ioc a b).map (addRightEmbedding c) = Ioc (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addRightEmbedding c) (Finset.Ioc a b) = Finset.Ioc (a + c) (b + c)
-- Finset.map_add_left_Ioo
@[simp] lemma map_add_left_Ioo (a b c : α) :
    (Ioo a b).map (addLeftEmbedding c) = Ioo (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addLeftEmbedding c) (Finset.Ioo a b) = Finset.Ioo (c + a) (c + b)
-- Finset.map_add_right_Ioo
@[simp] lemma map_add_right_Ioo (a b c : α) :
    (Ioo a b).map (addRightEmbedding c) = Ioo (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] (a b c : α)
Goal: Finset.map (addRightEmbedding c) (Finset.Ioo a b) = Finset.Ioo (a + c) (b + c)
-- Finset.image_add_left_Icc
@[simp] lemma image_add_left_Icc (a b c : α) : (Icc a b).image (c + ·) = Icc (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => c + x) (Finset.Icc a b) = Finset.Icc (c + a) (c + b)
-- Finset.image_add_left_Ico
@[simp] lemma image_add_left_Ico (a b c : α) : (Ico a b).image (c + ·) = Ico (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => c + x) (Finset.Ico a b) = Finset.Ico (c + a) (c + b)
-- Finset.image_add_left_Ioc
@[simp] lemma image_add_left_Ioc (a b c : α) : (Ioc a b).image (c + ·) = Ioc (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => c + x) (Finset.Ioc a b) = Finset.Ioc (c + a) (c + b)
-- Finset.image_add_left_Ioo
@[simp] lemma image_add_left_Ioo (a b c : α) : (Ioo a b).image (c + ·) = Ioo (c + a) (c + b) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => c + x) (Finset.Ioo a b) = Finset.Ioo (c + a) (c + b)
-- Finset.image_add_right_Icc
@[simp] lemma image_add_right_Icc (a b c : α) : (Icc a b).image (· + c) = Icc (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => x + c) (Finset.Icc a b) = Finset.Icc (a + c) (b + c)
-- Finset.image_add_right_Ico
@[simp] lemma image_add_right_Ico (a b c : α) : (Ico a b).image (· + c) = Ico (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => x + c) (Finset.Ico a b) = Finset.Ico (a + c) (b + c)
-- Finset.image_add_right_Ioc
@[simp] lemma image_add_right_Ioc (a b c : α) : (Ioc a b).image (· + c) = Ioc (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => x + c) (Finset.Ioc a b) = Finset.Ioc (a + c) (b + c)
-- Finset.image_add_right_Ioo
@[simp] lemma image_add_right_Ioo (a b c : α) : (Ioo a b).image (· + c) = Ioo (a + c) (b + c) 
Arguments: {α : Type u_2} [OrderedCancelAddCommMonoid α] [ExistsAddOfLE α] [LocallyFiniteOrder α] [DecidableEq α] (a b c : α)
Goal: Finset.image (fun x => x + c) (Finset.Ioo a b) = Finset.Ioo (a + c) (b + c)
-- OrderIso.mulLeft₀_apply
@[simps! (config := { simpRhs := true })]
def OrderIso.mulLeft₀ (a : α) (ha : 0 < a) : α ≃o α 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α)
Goal: (OrderIso.mulLeft₀ a ha) x = a * x
-- OrderIso.mulLeft₀_symm_apply
@[simps! (config := { simpRhs := true })]
def OrderIso.mulLeft₀ (a : α) (ha : 0 < a) : α ≃o α 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α)
Goal: (RelIso.symm (OrderIso.mulLeft₀ a ha)) x = a⁻¹ * x
-- OrderIso.mulRight₀_symm_apply
@[simps! (config := { simpRhs := true })]
def OrderIso.mulRight₀ (a : α) (ha : 0 < a) : α ≃o α 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α)
Goal: (RelIso.symm (OrderIso.mulRight₀ a ha)) x = x * a⁻¹
-- OrderIso.mulRight₀_apply
@[simps! (config := { simpRhs := true })]
def OrderIso.mulRight₀ (a : α) (ha : 0 < a) : α ≃o α 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] (a : α) (ha : 0 < a) (x : α)
Goal: (OrderIso.mulRight₀ a ha) x = x * a
-- le_div_iff
theorem le_div_iff (hc : 0 < c) : a ≤ b / c ↔ a * c ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a ≤ b / c ↔ a * c ≤ b
-- le_div_iff'
theorem le_div_iff' (hc : 0 < c) : a ≤ b / c ↔ c * a ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a ≤ b / c ↔ c * a ≤ b
-- div_le_iff
theorem div_le_iff (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 < b)
Goal: a / b ≤ c ↔ a ≤ c * b
-- div_le_iff'
theorem div_le_iff' (hb : 0 < b) : a / b ≤ c ↔ a ≤ b * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 < b)
Goal: a / b ≤ c ↔ a ≤ b * c
-- div_le_comm₀
lemma div_le_comm₀ (hb : 0 < b) (hc : 0 < c) : a / b ≤ c ↔ a / c ≤ b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hb : 0 < b) (hc : 0 < c)
Goal: a / b ≤ c ↔ a / c ≤ b
-- lt_div_iff
theorem lt_div_iff (hc : 0 < c) : a < b / c ↔ a * c < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a < b / c ↔ a * c < b
-- lt_div_iff'
theorem lt_div_iff' (hc : 0 < c) : a < b / c ↔ c * a < b 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: a < b / c ↔ c * a < b
-- div_lt_iff
theorem div_lt_iff (hc : 0 < c) : b / c < a ↔ b < a * c 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: b / c < a ↔ b < a * c
-- div_lt_iff'
theorem div_lt_iff' (hc : 0 < c) : b / c < a ↔ b < c * a 
Arguments: {α : Type u_2} [LinearOrderedSemifield α] {a b c : α} (hc : 0 < c)
Goal: b / c < a ↔ b < c * a