Documentation

Mathlib.Data.Set.Intervals.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 #

Instances for ↥(Set.Icc 0 1) #

instance Set.Icc.zero {α : Type u_1} [OrderedSemiring α] :
Zero (Set.Icc 0 1)
Equations
  • Set.Icc.zero = { zero := { val := 0, property := (_ : 0 Set.Icc 0 1) } }
instance Set.Icc.one {α : Type u_1} [OrderedSemiring α] :
One (Set.Icc 0 1)
Equations
  • Set.Icc.one = { one := { val := 1, property := (_ : 1 Set.Icc 0 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 Set.Icc 0 1) :
{ val := 0, property := h } = 0
@[simp]
theorem Set.Icc.mk_one {α : Type u_1} [OrderedSemiring α] (h : 1 Set.Icc 0 1) :
{ val := 1, property := h } = 1
@[simp]
theorem Set.Icc.coe_eq_zero {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} :
x = 0 x = 0
theorem Set.Icc.coe_ne_zero {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} :
x 0 x 0
@[simp]
theorem Set.Icc.coe_eq_one {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} :
x = 1 x = 1
theorem Set.Icc.coe_ne_one {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} :
x 1 x 1
theorem Set.Icc.coe_nonneg {α : Type u_1} [OrderedSemiring α] (x : (Set.Icc 0 1)) :
0 x
theorem Set.Icc.coe_le_one {α : Type u_1} [OrderedSemiring α] (x : (Set.Icc 0 1)) :
x 1
theorem Set.Icc.nonneg {α : Type u_1} [OrderedSemiring α] {t : (Set.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 : (Set.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 (Set.Icc 0 1)
Equations
  • Set.Icc.mul = { mul := fun (p q : (Set.Icc 0 1)) => { val := p * q, property := (_ : 0 p * q p * q 1) } }
instance Set.Icc.pow {α : Type u_1} [OrderedSemiring α] :
Pow (Set.Icc 0 1)
Equations
  • Set.Icc.pow = { pow := fun (p : (Set.Icc 0 1)) (n : ) => { val := p ^ n, property := (_ : 0 p ^ n p ^ n 1) } }
@[simp]
theorem Set.Icc.coe_mul {α : Type u_1} [OrderedSemiring α] (x : (Set.Icc 0 1)) (y : (Set.Icc 0 1)) :
(x * y) = x * y
@[simp]
theorem Set.Icc.coe_pow {α : Type u_1} [OrderedSemiring α] (x : (Set.Icc 0 1)) (n : ) :
(x ^ n) = x ^ n
theorem Set.Icc.mul_le_left {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} {y : (Set.Icc 0 1)} :
x * y x
theorem Set.Icc.mul_le_right {α : Type u_1} [OrderedSemiring α] {x : (Set.Icc 0 1)} {y : (Set.Icc 0 1)} :
x * y y
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
theorem Set.Icc.one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} (ht : t Set.Icc 0 1) :
1 - t Set.Icc 0 1
theorem Set.Icc.mem_iff_one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} :
t Set.Icc 0 1 1 - t Set.Icc 0 1
theorem Set.Icc.one_sub_nonneg {β : Type u_2} [OrderedRing β] (x : (Set.Icc 0 1)) :
0 1 - x
theorem Set.Icc.one_sub_le_one {β : Type u_2} [OrderedRing β] (x : (Set.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 (Set.Ico 0 1)
Equations
  • Set.Ico.zero = { zero := { val := 0, property := (_ : 0 Set.Ico 0 1) } }
@[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 Set.Ico 0 1) :
{ val := 0, property := h } = 0
@[simp]
theorem Set.Ico.coe_eq_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : (Set.Ico 0 1)} :
x = 0 x = 0
theorem Set.Ico.coe_ne_zero {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {x : (Set.Ico 0 1)} :
x 0 x 0
theorem Set.Ico.coe_nonneg {α : Type u_1} [OrderedSemiring α] (x : (Set.Ico 0 1)) :
0 x
theorem Set.Ico.coe_lt_one {α : Type u_1} [OrderedSemiring α] (x : (Set.Ico 0 1)) :
x < 1
theorem Set.Ico.nonneg {α : Type u_1} [OrderedSemiring α] [Nontrivial α] {t : (Set.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 (Set.Ico 0 1)
Equations
  • Set.Ico.mul = { mul := fun (p q : (Set.Ico 0 1)) => { val := p * q, property := (_ : 0 p * q p * q < 1) } }
@[simp]
theorem Set.Ico.coe_mul {α : Type u_1} [OrderedSemiring α] (x : (Set.Ico 0 1)) (y : (Set.Ico 0 1)) :
(x * y) = x * y
instance Set.Ico.semigroup {α : Type u_1} [OrderedSemiring α] :
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.

Instances for ↥(Set.Ioc 0 1) #

instance Set.Ioc.one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] :
One (Set.Ioc 0 1)
Equations
  • Set.Ioc.one = { one := { val := 1, property := (_ : 0 < 1 1 1) } }
@[simp]
theorem Set.Ioc.coe_one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] :
1 = 1
@[simp]
theorem Set.Ioc.mk_one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] (h : 1 Set.Ioc 0 1) :
{ val := 1, property := h } = 1
@[simp]
theorem Set.Ioc.coe_eq_one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] {x : (Set.Ioc 0 1)} :
x = 1 x = 1
theorem Set.Ioc.coe_ne_one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] {x : (Set.Ioc 0 1)} :
x 1 x 1
theorem Set.Ioc.coe_pos {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioc 0 1)) :
0 < x
theorem Set.Ioc.coe_le_one {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioc 0 1)) :
x 1
theorem Set.Ioc.le_one {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] {t : (Set.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 (Set.Ioc 0 1)
Equations
  • Set.Ioc.mul = { mul := fun (p q : (Set.Ioc 0 1)) => { val := p * q, property := (_ : 0 < p * q p * q 1) } }
instance Set.Ioc.pow {α : Type u_1} [StrictOrderedSemiring α] :
Pow (Set.Ioc 0 1)
Equations
  • Set.Ioc.pow = { pow := fun (p : (Set.Ioc 0 1)) (n : ) => { val := p ^ n, property := (_ : 0 < p ^ n p ^ n 1) } }
@[simp]
theorem Set.Ioc.coe_mul {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioc 0 1)) (y : (Set.Ioc 0 1)) :
(x * y) = x * y
@[simp]
theorem Set.Ioc.coe_pow {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioc 0 1)) (n : ) :
(x ^ n) = x ^ n
instance Set.Ioc.semigroup {α : Type u_1} [StrictOrderedSemiring α] :
Equations
  • One or more equations did not get rendered due to their size.
instance Set.Ioc.monoid {α : Type u_1} [StrictOrderedSemiring α] [Nontrivial α] :
Monoid (Set.Ioc 0 1)
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
Equations
Equations
  • Set.Ioc.cancelCommMonoid = let src := Set.Ioc.cancelMonoid; let src_1 := Set.Ioc.commMonoid; CancelCommMonoid.mk (_ : ∀ (a b : (Set.Ioc 0 1)), a * b = b * a)

Instances for ↥(Set.Ioo 0 1) #

theorem Set.Ioo.pos {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioo 0 1)) :
0 < x
theorem Set.Ioo.lt_one {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioo 0 1)) :
x < 1
instance Set.Ioo.mul {α : Type u_1} [StrictOrderedSemiring α] :
Mul (Set.Ioo 0 1)
Equations
  • Set.Ioo.mul = { mul := fun (p q : (Set.Ioo 0 1)) => { val := p * q, property := (_ : 0 < p * q p * q < 1) } }
@[simp]
theorem Set.Ioo.coe_mul {α : Type u_1} [StrictOrderedSemiring α] (x : (Set.Ioo 0 1)) (y : (Set.Ioo 0 1)) :
(x * y) = x * y
instance Set.Ioo.semigroup {α : Type u_1} [StrictOrderedSemiring α] :
Equations
  • One or more equations did not get rendered due to their size.
Equations
  • One or more equations did not get rendered due to their size.
theorem Set.Ioo.one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} (ht : t Set.Ioo 0 1) :
1 - t Set.Ioo 0 1
theorem Set.Ioo.mem_iff_one_sub_mem {β : Type u_2} [OrderedRing β] {t : β} :
t Set.Ioo 0 1 1 - t Set.Ioo 0 1
theorem Set.Ioo.one_minus_pos {β : Type u_2} [OrderedRing β] (x : (Set.Ioo 0 1)) :
0 < 1 - x
theorem Set.Ioo.one_minus_lt_one {β : Type u_2} [OrderedRing β] (x : (Set.Ioo 0 1)) :
1 - x < 1