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Mathlib.Algebra.Order.Interval.Set.Instances

Algebraic instances for unit intervals #

For suitably structured underlying type α, we exhibit the structure of the unit intervals (Set.Icc, Set.Ioc, Set.Ioc, and Set.Ioo) from 0 to 1. Note: Instances for the interval Ici 0 are dealt with in Algebra/Order/Nonneg.lean.

Main definitions #

The strongest typeclass provided on each interval is:

TODO #

algebraic instances for intervals -1 to 1
algebraic instances for Ici 1
algebraic instances for (Ioo (-1) 1)ᶜ
provide distribNeg instances where applicable
prove versions of mul_le_{left,right} for other intervals
prove versions of the lemmas in Topology/UnitInterval with ℝ generalized to some arbitrary ordered semiring

Instances for `↥(Set.Icc 0 1)` #

instance Set.Icc.zero {α : Type u_1} [OrderedSemiring α] :

Zero ↑(Icc 0 1)

Equations

Set.Icc.zero = { zero := ⟨0, ⋯⟩ }

instance Set.Icc.one {α : Type u_1} [OrderedSemiring α] :

One ↑(Icc 0 1)

Equations

Set.Icc.one = { one := ⟨1, ⋯⟩ }

@[simp]

theorem Set.Icc.coe_zero {α : Type u_1} [OrderedSemiring α] :

↑0 = 0

@[simp]

theorem Set.Icc.coe_one {α : Type u_1} [OrderedSemiring α] :

↑1 = 1

@[simp]

theorem Set.Icc.mk_zero {α : Type u_1} [OrderedSemiring α] (h : 0 ∈ Icc 0 1) :

⟨0, h⟩ = 0

@[simp]

theorem Set.Icc.mk_one {α : Type u_1} [OrderedSemiring α] (h : 1 ∈ Icc 0 1) :

⟨1, h⟩ = 1

@[simp]

theorem Set.Icc.coe_eq_zero {α : Type u_1} [OrderedSemiring α] {x : ↑(Icc 0 1)} :

↑x = 0 ↔ x = 0

theorem Set.Icc.coe_ne_zero {α : Type u_1} [OrderedSemiring α] {x : ↑(Icc 0 1)} :

↑x ≠ 0 ↔ x ≠ 0

@[simp]

theorem Set.Icc.coe_eq_one {α : Type u_1} [OrderedSemiring α] {x : ↑(Icc 0 1)} :

↑x = 1 ↔ x = 1

theorem Set.Icc.coe_ne_one {α : Type u_1} [OrderedSemiring α] {x : ↑(Icc 0 1)} :

↑x ≠ 1 ↔ x ≠ 1

theorem Set.Icc.coe_nonneg {α : Type u_1} [OrderedSemiring α] (x : ↑(Icc 0 1)) :

0 ≤ ↑x

theorem Set.Icc.coe_le_one {α : Type u_1} [OrderedSemiring α] (x : ↑(Icc 0 1)) :

↑x ≤ 1

theorem Set.Icc.nonneg {α : Type u_1} [OrderedSemiring α] {t : ↑(Icc 0 1)} :

0 ≤ t

like coe_nonneg, but with the inequality in Icc (0:α) 1.

theorem Set.Icc.le_one {α : Type u_1} [OrderedSemiring α] {t : ↑(Icc 0 1)} :

t ≤ 1

like coe_le_one, but with the inequality in Icc (0:α) 1.

instance Set.Icc.mul {α : Type u_1} [OrderedSemiring α] :

Mul ↑(Icc 0 1)

Equations

Set.Icc.mul = { mul := fun (p q : ↑(Set.Icc 0 1)) => ⟨↑p * ↑q, ⋯⟩ }

instance Set.Icc.pow {α : Type u_1} [OrderedSemiring α] :

Pow ↑(Icc 0 1) ℕ

Equations

Set.Icc.pow = { pow := fun (p : ↑(Set.Icc 0 1)) (n : ℕ) => ⟨↑p ^ n, ⋯⟩ }

@[simp]

theorem Set.Icc.coe_mul {α : Type u_1} [OrderedSemiring α] (x y : ↑(Icc 0 1)) :

↑(x * y) = ↑x * ↑y

@[simp]

theorem Set.Icc.coe_pow {α : Type u_1} [OrderedSemiring α] (x : ↑(Icc 0 1)) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

theorem Set.Icc.mul_le_left {α : Type u_1} [OrderedSemiring α] {x y : ↑(Icc 0 1)} :

x * y ≤ x

theorem Set.Icc.mul_le_right {α : Type u_1} [OrderedSemiring α] {x y : ↑(Icc 0 1)} :

x * y ≤ y

instance Set.Icc.monoidWithZero {α : Type u_1} [OrderedSemiring α] :

MonoidWithZero ↑(Icc 0 1)

Equations

One or more equations did not get rendered due to their size.

instance Set.Icc.commMonoidWithZero {α : Type u_2} [OrderedCommSemiring α] :

CommMonoidWithZero ↑(Icc 0 1)

Equations

Set.Icc.commMonoidWithZero = { toMonoid := MonoidWithZero.toMonoid, mul_comm := ⋯, toZero := Set.Icc.zero, zero_mul := ⋯, mul_zero := ⋯ }

instance Set.Icc.cancelMonoidWithZero {α : Type u_2} [OrderedRing α] [NoZeroDivisors α] :

CancelMonoidWithZero ↑(Icc 0 1)

Equations

Set.Icc.cancelMonoidWithZero = { toMonoidWithZero := Set.Icc.monoidWithZero, toIsCancelMulZero := ⋯ }

instance Set.Icc.cancelCommMonoidWithZero {α : Type u_2} [OrderedCommRing α] [NoZeroDivisors α] :

CancelCommMonoidWithZero ↑(Icc 0 1)

Equations

Set.Icc.cancelCommMonoidWithZero = { toCommMonoidWithZero := Set.Icc.commMonoidWithZero, toIsLeftCancelMulZero := ⋯ }

theorem Set.Icc.one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} (ht : t ∈ Icc 0 1) :

1 - t ∈ Icc 0 1

theorem Set.Icc.mem_iff_one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} :

t ∈ Icc 0 1 ↔ 1 - t ∈ Icc 0 1

theorem Set.Icc.one_sub_nonneg {β : Type u_2} [OrderedRing β] (x : ↑(Icc 0 1)) :

0 ≤ 1 - ↑x

theorem Set.Icc.one_sub_le_one {β : Type u_2} [OrderedRing β] (x : ↑(Icc 0 1)) :

1 - ↑x ≤ 1

Instances for `↥(Set.Ico 0 1)` #

instance Set.Ico.zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] :

Zero ↑(Ico 0 1)

Equations

Set.Ico.zero = { zero := ⟨0, ⋯⟩ }

@[simp]

theorem Set.Ico.coe_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] :

↑0 = 0

@[simp]

theorem Set.Ico.mk_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] (h : 0 ∈ Ico 0 1) :

⟨0, h⟩ = 0

@[simp]

theorem Set.Ico.coe_eq_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : ↑(Ico 0 1)} :

↑x = 0 ↔ x = 0

theorem Set.Ico.coe_ne_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : ↑(Ico 0 1)} :

↑x ≠ 0 ↔ x ≠ 0

theorem Set.Ico.coe_nonneg {α : Type u_1} [OrderedSemiring α] (x : ↑(Ico 0 1)) :

0 ≤ ↑x

theorem Set.Ico.coe_lt_one {α : Type u_1} [OrderedSemiring α] (x : ↑(Ico 0 1)) :

↑x < 1

theorem Set.Ico.nonneg {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {t : ↑(Ico 0 1)} :

0 ≤ t

like coe_nonneg, but with the inequality in Ico (0:α) 1.

instance Set.Ico.mul {α : Type u_1} [OrderedSemiring α] :

Mul ↑(Ico 0 1)

Equations

Set.Ico.mul = { mul := fun (p q : ↑(Set.Ico 0 1)) => ⟨↑p * ↑q, ⋯⟩ }

@[simp]

theorem Set.Ico.coe_mul {α : Type u_1} [OrderedSemiring α] (x y : ↑(Ico 0 1)) :

↑(x * y) = ↑x * ↑y

instance Set.Ico.semigroup {α : Type u_1} [OrderedSemiring α] :

Semigroup ↑(Ico 0 1)

Equations

Set.Ico.semigroup = { toMul := Set.Ico.mul, mul_assoc := ⋯ }

instance Set.Ico.commSemigroup {α : Type u_2} [OrderedCommSemiring α] :

CommSemigroup ↑(Ico 0 1)

Equations

Set.Ico.commSemigroup = { toSemigroup := Set.Ico.semigroup, mul_comm := ⋯ }

Instances for `↥(Set.Ioc 0 1)` #

instance Set.Ioc.one {α : Type u_1} [StrictOrderedSemiring α] :

One ↑(Ioc 0 1)

Equations

Set.Ioc.one = { one := ⟨1, ⋯⟩ }

@[simp]

theorem Set.Ioc.coe_one {α : Type u_1} [StrictOrderedSemiring α] :

↑1 = 1

@[simp]

theorem Set.Ioc.mk_one {α : Type u_1} [StrictOrderedSemiring α] (h : 1 ∈ Ioc 0 1) :

⟨1, h⟩ = 1

@[simp]

theorem Set.Ioc.coe_eq_one {α : Type u_1} [StrictOrderedSemiring α] {x : ↑(Ioc 0 1)} :

↑x = 1 ↔ x = 1

theorem Set.Ioc.coe_ne_one {α : Type u_1} [StrictOrderedSemiring α] {x : ↑(Ioc 0 1)} :

↑x ≠ 1 ↔ x ≠ 1

theorem Set.Ioc.coe_pos {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Ioc 0 1)) :

0 < ↑x

theorem Set.Ioc.coe_le_one {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Ioc 0 1)) :

↑x ≤ 1

theorem Set.Ioc.le_one {α : Type u_1} [StrictOrderedSemiring α] {t : ↑(Ioc 0 1)} :

t ≤ 1

like coe_le_one, but with the inequality in Ioc (0:α) 1.

instance Set.Ioc.mul {α : Type u_1} [StrictOrderedSemiring α] :

Mul ↑(Ioc 0 1)

Equations

Set.Ioc.mul = { mul := fun (p q : ↑(Set.Ioc 0 1)) => ⟨↑p * ↑q, ⋯⟩ }

instance Set.Ioc.pow {α : Type u_1} [StrictOrderedSemiring α] :

Pow ↑(Ioc 0 1) ℕ

Equations

Set.Ioc.pow = { pow := fun (p : ↑(Set.Ioc 0 1)) (n : ℕ) => ⟨↑p ^ n, ⋯⟩ }

@[simp]

theorem Set.Ioc.coe_mul {α : Type u_1} [StrictOrderedSemiring α] (x y : ↑(Ioc 0 1)) :

↑(x * y) = ↑x * ↑y

@[simp]

theorem Set.Ioc.coe_pow {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Ioc 0 1)) (n : ℕ) :

↑(x ^ n) = ↑x ^ n

instance Set.Ioc.semigroup {α : Type u_1} [StrictOrderedSemiring α] :

Semigroup ↑(Ioc 0 1)

Equations

Set.Ioc.semigroup = { toMul := Set.Ioc.mul, mul_assoc := ⋯ }

instance Set.Ioc.monoid {α : Type u_1} [StrictOrderedSemiring α] :

Monoid ↑(Ioc 0 1)

Equations

One or more equations did not get rendered due to their size.

instance Set.Ioc.commSemigroup {α : Type u_2} [StrictOrderedCommSemiring α] :

CommSemigroup ↑(Ioc 0 1)

Equations

Set.Ioc.commSemigroup = { toSemigroup := Set.Ioc.semigroup, mul_comm := ⋯ }

instance Set.Ioc.commMonoid {α : Type u_2} [StrictOrderedCommSemiring α] :

CommMonoid ↑(Ioc 0 1)

Equations

Set.Ioc.commMonoid = { toMonoid := Set.Ioc.monoid, mul_comm := ⋯ }

instance Set.Ioc.cancelMonoid {α : Type u_2} [StrictOrderedRing α] [IsDomain α] :

CancelMonoid ↑(Ioc 0 1)

Equations

Set.Ioc.cancelMonoid = { toMonoid := Set.Ioc.monoid, mul_left_cancel := ⋯, mul_right_cancel := ⋯ }

instance Set.Ioc.cancelCommMonoid {α : Type u_2} [StrictOrderedCommRing α] [IsDomain α] :

CancelCommMonoid ↑(Ioc 0 1)

Equations

Set.Ioc.cancelCommMonoid = { toMonoid := LeftCancelMonoid.toMonoid, mul_comm := ⋯, mul_left_cancel := ⋯ }

Instances for `↥(Set.Ioo 0 1)` #

theorem Set.Ioo.pos {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Ioo 0 1)) :

0 < ↑x

theorem Set.Ioo.lt_one {α : Type u_1} [StrictOrderedSemiring α] (x : ↑(Ioo 0 1)) :

↑x < 1

instance Set.Ioo.mul {α : Type u_1} [StrictOrderedSemiring α] :

Mul ↑(Ioo 0 1)

Equations

Set.Ioo.mul = { mul := fun (p q : ↑(Set.Ioo 0 1)) => ⟨↑p * ↑q, ⋯⟩ }

@[simp]

theorem Set.Ioo.coe_mul {α : Type u_1} [StrictOrderedSemiring α] (x y : ↑(Ioo 0 1)) :

↑(x * y) = ↑x * ↑y

instance Set.Ioo.semigroup {α : Type u_1} [StrictOrderedSemiring α] :

Semigroup ↑(Ioo 0 1)

Equations

Set.Ioo.semigroup = { toMul := Set.Ioo.mul, mul_assoc := ⋯ }

instance Set.Ioo.commSemigroup {α : Type u_2} [StrictOrderedCommSemiring α] :

CommSemigroup ↑(Ioo 0 1)

Equations

Set.Ioo.commSemigroup = { toSemigroup := Set.Ioo.semigroup, mul_comm := ⋯ }

theorem Set.Ioo.one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} (ht : t ∈ Ioo 0 1) :

1 - t ∈ Ioo 0 1

theorem Set.Ioo.mem_iff_one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} :

t ∈ Ioo 0 1 ↔ 1 - t ∈ Ioo 0 1

theorem Set.Ioo.one_minus_pos {β : Type u_2} [OrderedRing β] (x : ↑(Ioo 0 1)) :

0 < 1 - ↑x

theorem Set.Ioo.one_minus_lt_one {β : Type u_2} [OrderedRing β] (x : ↑(Ioo 0 1)) :

1 - ↑x < 1