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

full_name: Set.Icc.zero
code: instance zero : Zero (Icc (0 : α) 1) 
goal: Zero ↑(Set.Icc 0 1)

full_name: Set.Icc.one
code: instance one : One (Icc (0 : α) 1) 
goal: One ↑(Set.Icc 0 1)

full_name: Set.Icc.coe_zero
code: @[simp, norm_cast]
theorem coe_zero : ↑(0 : Icc (0 : α) 1) = (0 : α) 
goal: ↑0 = 0

full_name: Set.Icc.coe_one
code: @[simp, norm_cast]
theorem coe_one : ↑(1 : Icc (0 : α) 1) = (1 : α) 
goal: ↑1 = 1

full_name: Set.Icc.mk_zero
code: @[simp]
theorem mk_zero (h : (0 : α) ∈ Icc (0 : α) 1) : (⟨0, h⟩ : Icc (0 : α) 1) = 0 
goal: ⟨0, h⟩ = 0

full_name: Set.Icc.mk_one
code: @[simp]
theorem mk_one (h : (1 : α) ∈ Icc (0 : α) 1) : (⟨1, h⟩ : Icc (0 : α) 1) = 1 
goal: ⟨1, h⟩ = 1

full_name: Set.Icc.coe_eq_zero
code: @[simp, norm_cast]
theorem coe_eq_zero {x : Icc (0 : α) 1} : (x : α) = 0 ↔ x = 0 
goal: ↑x = 0 ↔ x = 0

full_name: Set.Icc.coe_ne_zero
code: theorem coe_ne_zero {x : Icc (0 : α) 1} : (x : α) ≠ 0 ↔ x ≠ 0 
goal: ↑x ≠ 0 ↔ x ≠ 0

full_name: Set.Icc.coe_eq_one
code: @[simp, norm_cast]
theorem coe_eq_one {x : Icc (0 : α) 1} : (x : α) = 1 ↔ x = 1 
goal: ↑x = 1 ↔ x = 1

full_name: Set.Icc.coe_ne_one
code: theorem coe_ne_one {x : Icc (0 : α) 1} : (x : α) ≠ 1 ↔ x ≠ 1 
goal: ↑x ≠ 1 ↔ x ≠ 1

full_name: Set.Icc.coe_nonneg
code: theorem coe_nonneg (x : Icc (0 : α) 1) : 0 ≤ (x : α) 
goal: 0 ≤ ↑x

full_name: Set.Icc.coe_le_one
code: theorem coe_le_one (x : Icc (0 : α) 1) : (x : α) ≤ 1 
goal: ↑x ≤ 1

full_name: Set.Icc.nonneg
code: /-- like `coe_nonneg`, but with the inequality in `Icc (0:α) 1`. -/
theorem nonneg {t : Icc (0 : α) 1} : 0 ≤ t 
goal: 0 ≤ t

full_name: Set.Icc.le_one
code: /-- like `coe_le_one`, but with the inequality in `Icc (0:α) 1`. -/
theorem le_one {t : Icc (0 : α) 1} : t ≤ 1 
goal: t ≤ 1

full_name: Set.Icc.mul
code: instance mul : Mul (Icc (0 : α) 1) 
goal: Mul ↑(Set.Icc 0 1)

full_name: Set.Icc.pow
code: instance pow : Pow (Icc (0 : α) 1) ℕ 
goal: Pow ↑(Set.Icc 0 1) ℕ

full_name: Set.Icc.coe_mul
code: @[simp, norm_cast]
theorem coe_mul (x y : Icc (0 : α) 1) : ↑(x * y) = (x * y : α) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Set.Icc.coe_pow
code: @[simp, norm_cast]
theorem coe_pow (x : Icc (0 : α) 1) (n : ℕ) : ↑(x ^ n) = ((x : α) ^ n) 
goal: ↑(x ^ n) = ↑x ^ n

full_name: Set.Icc.mul_le_left
code: theorem mul_le_left {x y : Icc (0 : α) 1} : x * y ≤ x 
goal: x * y ≤ x

full_name: Set.Icc.mul_le_right
code: theorem mul_le_right {x y : Icc (0 : α) 1} : x * y ≤ y 
goal: x * y ≤ y

full_name: Set.Icc.monoidWithZero
code: instance monoidWithZero : MonoidWithZero (Icc (0 : α) 1) 
goal: MonoidWithZero ↑(Set.Icc 0 1)

full_name: Set.Icc.commMonoidWithZero
code: instance commMonoidWithZero {α : Type*} [OrderedCommSemiring α] :
    CommMonoidWithZero (Icc (0 : α) 1) 
goal: CommMonoidWithZero ↑(Set.Icc 0 1)

full_name: Set.Icc.cancelMonoidWithZero
code: instance cancelMonoidWithZero {α : Type*} [OrderedRing α] [NoZeroDivisors α] :
    CancelMonoidWithZero (Icc (0 : α) 1) 
goal: CancelMonoidWithZero ↑(Set.Icc 0 1)

full_name: Set.Icc.cancelCommMonoidWithZero
code: instance cancelCommMonoidWithZero {α : Type*} [OrderedCommRing α] [NoZeroDivisors α] :
    CancelCommMonoidWithZero (Icc (0 : α) 1) 
goal: CancelCommMonoidWithZero ↑(Set.Icc 0 1)

full_name: Set.Icc.one_sub_mem
code: theorem one_sub_mem {t : β} (ht : t ∈ Icc (0 : β) 1) : 1 - t ∈ Icc (0 : β) 1 
goal: 1 - t ∈ Set.Icc 0 1

full_name: Set.Icc.mem_iff_one_sub_mem
code: theorem mem_iff_one_sub_mem {t : β} : t ∈ Icc (0 : β) 1 ↔ 1 - t ∈ Icc (0 : β) 1 
goal: t ∈ Set.Icc 0 1 ↔ 1 - t ∈ Set.Icc 0 1

full_name: Set.Icc.one_sub_nonneg
code: theorem one_sub_nonneg (x : Icc (0 : β) 1) : 0 ≤ 1 - (x : β) 
goal: 0 ≤ 1 - ↑x

full_name: Set.Icc.one_sub_le_one
code: theorem one_sub_le_one (x : Icc (0 : β) 1) : 1 - (x : β) ≤ 1 
goal: 1 - ↑x ≤ 1

full_name: Set.Ico.zero
code: instance zero [Nontrivial α] : Zero (Ico (0 : α) 1) 
goal: Zero ↑(Set.Ico 0 1)

full_name: Set.Ico.coe_zero
code: @[simp, norm_cast]
theorem coe_zero [Nontrivial α] : ↑(0 : Ico (0 : α) 1) = (0 : α) 
goal: ↑0 = 0

full_name: Set.Ico.mk_zero
code: @[simp]
theorem mk_zero [Nontrivial α] (h : (0 : α) ∈ Ico (0 : α) 1) : (⟨0, h⟩ : Ico (0 : α) 1) = 0 
goal: ⟨0, h⟩ = 0

full_name: Set.Ico.coe_eq_zero
code: @[simp, norm_cast]
theorem coe_eq_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) = 0 ↔ x = 0 
goal: ↑x = 0 ↔ x = 0

full_name: Set.Ico.coe_ne_zero
code: theorem coe_ne_zero [Nontrivial α] {x : Ico (0 : α) 1} : (x : α) ≠ 0 ↔ x ≠ 0 
goal: ↑x ≠ 0 ↔ x ≠ 0

full_name: Set.Ico.coe_nonneg
code: theorem coe_nonneg (x : Ico (0 : α) 1) : 0 ≤ (x : α) 
goal: 0 ≤ ↑x

full_name: Set.Ico.coe_lt_one
code: theorem coe_lt_one (x : Ico (0 : α) 1) : (x : α) < 1 
goal: ↑x < 1

full_name: Set.Ico.nonneg
code: /-- like `coe_nonneg`, but with the inequality in `Ico (0:α) 1`. -/
theorem nonneg [Nontrivial α] {t : Ico (0 : α) 1} : 0 ≤ t 
goal: 0 ≤ t

full_name: Set.Ico.mul
code: instance mul : Mul (Ico (0 : α) 1) 
goal: Mul ↑(Set.Ico 0 1)

full_name: Set.Ico.coe_mul
code: @[simp, norm_cast]
theorem coe_mul (x y : Ico (0 : α) 1) : ↑(x * y) = (x * y : α) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Set.Ico.semigroup
code: instance semigroup : Semigroup (Ico (0 : α) 1) 
goal: Semigroup ↑(Set.Ico 0 1)

full_name: Set.Ico.commSemigroup
code: instance commSemigroup {α : Type*} [OrderedCommSemiring α] : CommSemigroup (Ico (0 : α) 1) 
goal: CommSemigroup ↑(Set.Ico 0 1)

full_name: Set.Ioc.one
code: instance one : One (Ioc (0 : α) 1) 
goal: One ↑(Set.Ioc 0 1)

full_name: Set.Ioc.coe_one
code: @[simp, norm_cast]
theorem coe_one : ↑(1 : Ioc (0 : α) 1) = (1 : α) 
goal: ↑1 = 1

full_name: Set.Ioc.mk_one
code: @[simp]
theorem mk_one (h : (1 : α) ∈ Ioc (0 : α) 1) : (⟨1, h⟩ : Ioc (0 : α) 1) = 1 
goal: ⟨1, h⟩ = 1

full_name: Set.Ioc.coe_eq_one
code: @[simp, norm_cast]
theorem coe_eq_one {x : Ioc (0 : α) 1} : (x : α) = 1 ↔ x = 1 
goal: ↑x = 1 ↔ x = 1

full_name: Set.Ioc.coe_ne_one
code: theorem coe_ne_one {x : Ioc (0 : α) 1} : (x : α) ≠ 1 ↔ x ≠ 1 
goal: ↑x ≠ 1 ↔ x ≠ 1

full_name: Set.Ioc.coe_pos
code: theorem coe_pos (x : Ioc (0 : α) 1) : 0 < (x : α) 
goal: 0 < ↑x

full_name: Set.Ioc.coe_le_one
code: theorem coe_le_one (x : Ioc (0 : α) 1) : (x : α) ≤ 1 
goal: ↑x ≤ 1

full_name: Set.Ioc.le_one
code: /-- like `coe_le_one`, but with the inequality in `Ioc (0:α) 1`. -/
theorem le_one {t : Ioc (0 : α) 1} : t ≤ 1 
goal: t ≤ 1

full_name: Set.Ioc.mul
code: instance mul : Mul (Ioc (0 : α) 1) 
goal: Mul ↑(Set.Ioc 0 1)

full_name: Set.Ioc.pow
code: instance pow : Pow (Ioc (0 : α) 1) ℕ 
goal: Pow ↑(Set.Ioc 0 1) ℕ

full_name: Set.Ioc.coe_mul
code: @[simp, norm_cast]
theorem coe_mul (x y : Ioc (0 : α) 1) : ↑(x * y) = (x * y : α) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Set.Ioc.coe_pow
code: @[simp, norm_cast]
theorem coe_pow (x : Ioc (0 : α) 1) (n : ℕ) : ↑(x ^ n) = ((x : α) ^ n) 
goal: ↑(x ^ n) = ↑x ^ n

full_name: Set.Ioc.semigroup
code: instance semigroup : Semigroup (Ioc (0 : α) 1) 
goal: Semigroup ↑(Set.Ioc 0 1)

full_name: Set.Ioc.monoid
code: instance monoid : Monoid (Ioc (0 : α) 1) 
goal: Monoid ↑(Set.Ioc 0 1)

full_name: Set.Ioc.commSemigroup
code: instance commSemigroup {α : Type*} [StrictOrderedCommSemiring α] : CommSemigroup (Ioc (0 : α) 1) 
goal: CommSemigroup ↑(Set.Ioc 0 1)

full_name: Set.Ioc.commMonoid
code: instance commMonoid {α : Type*} [StrictOrderedCommSemiring α] :
    CommMonoid (Ioc (0 : α) 1) 
goal: CommMonoid ↑(Set.Ioc 0 1)

full_name: Set.Ioc.cancelMonoid
code: instance cancelMonoid {α : Type*} [StrictOrderedRing α] [IsDomain α] :
    CancelMonoid (Ioc (0 : α) 1) 
goal: CancelMonoid ↑(Set.Ioc 0 1)

full_name: Set.Ioc.cancelCommMonoid
code: instance cancelCommMonoid {α : Type*} [StrictOrderedCommRing α] [IsDomain α] :
    CancelCommMonoid (Ioc (0 : α) 1) 
goal: CancelCommMonoid ↑(Set.Ioc 0 1)

full_name: Set.Ioo.pos
code: theorem pos (x : Ioo (0 : α) 1) : 0 < (x : α) 
goal: 0 < ↑x

full_name: Set.Ioo.lt_one
code: theorem lt_one (x : Ioo (0 : α) 1) : (x : α) < 1 
goal: ↑x < 1

full_name: Set.Ioo.mul
code: instance mul : Mul (Ioo (0 : α) 1) 
goal: Mul ↑(Set.Ioo 0 1)

full_name: Set.Ioo.coe_mul
code: @[simp, norm_cast]
theorem coe_mul (x y : Ioo (0 : α) 1) : ↑(x * y) = (x * y : α) 
goal: ↑(x * y) = ↑x * ↑y

full_name: Set.Ioo.semigroup
code: instance semigroup : Semigroup (Ioo (0 : α) 1) 
goal: Semigroup ↑(Set.Ioo 0 1)

full_name: Set.Ioo.commSemigroup
code: instance commSemigroup {α : Type*} [StrictOrderedCommSemiring α] : CommSemigroup (Ioo (0 : α) 1) 
goal: CommSemigroup ↑(Set.Ioo 0 1)

full_name: Set.Ioo.one_sub_mem
code: theorem one_sub_mem {t : β} (ht : t ∈ Ioo (0 : β) 1) : 1 - t ∈ Ioo (0 : β) 1 
goal: 1 - t ∈ Set.Ioo 0 1

full_name: Set.Ioo.mem_iff_one_sub_mem
code: theorem mem_iff_one_sub_mem {t : β} : t ∈ Ioo (0 : β) 1 ↔ 1 - t ∈ Ioo (0 : β) 1 
goal: t ∈ Set.Ioo 0 1 ↔ 1 - t ∈ Set.Ioo 0 1

full_name: Set.Ioo.one_minus_pos
code: theorem one_minus_pos (x : Ioo (0 : β) 1) : 0 < 1 - (x : β) 
goal: 0 < 1 - ↑x

full_name: Set.Ioo.one_minus_lt_one
code: theorem one_minus_lt_one (x : Ioo (0 : β) 1) : 1 - (x : β) < 1 
goal: 1 - ↑x < 1