Interval arithmetic

This file defines arithmetic operations on intervals and prove their correctness. Note that this is full precision operations. The essentials of float operations can be found in Data.FP.Basic. We have not yet integrated these with the rest of the library.

One/zero

instance instOneNonemptyInterval {α : Type u_2} [Preorder α] [One α] : One (NonemptyInterval α)

Equations

• instOneNonemptyInterval = { one := NonemptyInterval.pure 1 }

instance instZeroNonemptyInterval {α : Type u_2} [Preorder α] [Zero α] : Zero (NonemptyInterval α)

Equations

• instZeroNonemptyInterval = { zero := NonemptyInterval.pure 0 }

@[simp]

theorem NonemptyInterval.toProd_one {α : Type u_2} [Preorder α] [One α] : NonemptyInterval.toProd 1 = 1

@[simp]

theorem NonemptyInterval.toProd_zero {α : Type u_2} [Preorder α] [Zero α] : NonemptyInterval.toProd 0 = 0

theorem NonemptyInterval.fst_one {α : Type u_2} [Preorder α] [One α] : (NonemptyInterval.toProd 1).1 = 1

theorem NonemptyInterval.fst_zero {α : Type u_2} [Preorder α] [Zero α] : (NonemptyInterval.toProd 0).1 = 0

theorem NonemptyInterval.snd_one {α : Type u_2} [Preorder α] [One α] : (NonemptyInterval.toProd 1).2 = 1

theorem NonemptyInterval.snd_zero {α : Type u_2} [Preorder α] [Zero α] : (NonemptyInterval.toProd 0).2 = 0

@[simp]

theorem NonemptyInterval.coe_one_interval {α : Type u_2} [Preorder α] [One α] : ↑1 = 1

@[simp]

theorem NonemptyInterval.coe_zero_interval {α : Type u_2} [Preorder α] [Zero α] : ↑0 = 0

@[simp]

theorem NonemptyInterval.pure_one {α : Type u_2} [Preorder α] [One α] : NonemptyInterval.pure 1 = 1

@[simp]

theorem NonemptyInterval.pure_zero {α : Type u_2} [Preorder α] [Zero α] : NonemptyInterval.pure 0 = 0

@[simp]

theorem Interval.pure_one {α : Type u_2} [Preorder α] [One α] : Interval.pure 1 = 1

@[simp]

theorem Interval.pure_zero {α : Type u_2} [Preorder α] [Zero α] : Interval.pure 0 = 0

theorem Interval.one_ne_bot {α : Type u_2} [Preorder α] [One α] : 1 ≠ ⊥

theorem Interval.zero_ne_bot {α : Type u_2} [Preorder α] [Zero α] : 0 ≠ ⊥

theorem Interval.bot_ne_one {α : Type u_2} [Preorder α] [One α] : ⊥ ≠ 1

theorem Interval.bot_ne_zero {α : Type u_2} [Preorder α] [Zero α] : ⊥ ≠ 0

@[simp]

theorem NonemptyInterval.coe_one {α : Type u_2} [PartialOrder α] [One α] : ↑1 = 1

@[simp]

theorem NonemptyInterval.coe_zero {α : Type u_2} [PartialOrder α] [Zero α] : ↑0 = 0

theorem NonemptyInterval.one_mem_one {α : Type u_2} [PartialOrder α] [One α] : 1 ∈ 1

theorem NonemptyInterval.zero_mem_zero {α : Type u_2} [PartialOrder α] [Zero α] : 0 ∈ 0

@[simp]

theorem Interval.coe_one {α : Type u_2} [PartialOrder α] [One α] : ↑1 = 1

@[simp]

theorem Interval.coe_zero {α : Type u_2} [PartialOrder α] [Zero α] : ↑0 = 0

theorem Interval.one_mem_one {α : Type u_2} [PartialOrder α] [One α] : 1 ∈ 1

theorem Interval.zero_mem_zero {α : Type u_2} [PartialOrder α] [Zero α] : 0 ∈ 0

Addition/multiplication

Note that this multiplication does not apply to ℚ or ℝ.

instance instMulNonemptyInterval {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] : Mul (NonemptyInterval α)

Equations

• instMulNonemptyInterval = { mul := fun (s t : NonemptyInterval α) => { toProd := s.toProd * t.toProd, fst_le_snd := ⋯ } }

instance instAddNonemptyInterval {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] : Add (NonemptyInterval α)

Equations

• instAddNonemptyInterval = { add := fun (s t : NonemptyInterval α) => { toProd := s.toProd + t.toProd, fst_le_snd := ⋯ } }

instance instMulInterval {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] : Mul (Interval α)

Equations

• instMulInterval = { mul := Option.map₂ fun (x1 x2 : NonemptyInterval α) => x1 * x2 }

instance instAddInterval {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] : Add (Interval α)

Equations

• instAddInterval = { add := Option.map₂ fun (x1 x2 : NonemptyInterval α) => x1 + x2 }

@[simp]

theorem NonemptyInterval.toProd_mul {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s t : NonemptyInterval α) : (s * t).toProd = s.toProd * t.toProd

@[simp]

theorem NonemptyInterval.toProd_add {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : (s + t).toProd = s.toProd + t.toProd

theorem NonemptyInterval.fst_mul {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s t : NonemptyInterval α) : (s * t).toProd.1 = s.toProd.1 * t.toProd.1

theorem NonemptyInterval.fst_add {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : (s + t).toProd.1 = s.toProd.1 + t.toProd.1

theorem NonemptyInterval.snd_mul {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s t : NonemptyInterval α) : (s * t).toProd.2 = s.toProd.2 * t.toProd.2

theorem NonemptyInterval.snd_add {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : (s + t).toProd.2 = s.toProd.2 + t.toProd.2

@[simp]

theorem NonemptyInterval.coe_mul_interval {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s t : NonemptyInterval α) : ↑(s * t) = ↑s * ↑t

@[simp]

theorem NonemptyInterval.coe_add_interval {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s t : NonemptyInterval α) : ↑(s + t) = ↑s + ↑t

@[simp]

theorem NonemptyInterval.pure_mul_pure {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (a b : α) : NonemptyInterval.pure a * NonemptyInterval.pure b = NonemptyInterval.pure (a * b)

@[simp]

theorem NonemptyInterval.pure_add_pure {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (a b : α) : NonemptyInterval.pure a + NonemptyInterval.pure b = NonemptyInterval.pure (a + b)

@[simp]

theorem Interval.bot_mul {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (t : Interval α) : ⊥ * t = ⊥

@[simp]

theorem Interval.bot_add {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (t : Interval α) : ⊥ + t = ⊥

@[simp]

theorem Interval.mul_bot {α : Type u_2} [Preorder α] [Mul α] [MulLeftMono α] [MulRightMono α] (s : Interval α) : s * ⊥ = ⊥

theorem Interval.add_bot {α : Type u_2} [Preorder α] [Add α] [AddLeftMono α] [AddRightMono α] (s : Interval α) : s + ⊥ = ⊥

Powers

instance NonemptyInterval.hasNSMul {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] : SMul ℕ (NonemptyInterval α)

Equations

• NonemptyInterval.hasNSMul = { smul := fun (n : ℕ) (s : NonemptyInterval α) => { toProd := (n • s.toProd.1, n • s.toProd.2), fst_le_snd := ⋯ } }

instance NonemptyInterval.hasPow {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] : Pow (NonemptyInterval α) ℕ

Equations

• NonemptyInterval.hasPow = { pow := fun (s : NonemptyInterval α) (n : ℕ) => { toProd := s.toProd ^ n, fst_le_snd := ⋯ } }

@[simp]

theorem NonemptyInterval.toProd_pow {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd = s.toProd ^ n

@[simp]

theorem NonemptyInterval.toProd_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd = n • s.toProd

theorem NonemptyInterval.fst_pow {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd.1 = s.toProd.1 ^ n

theorem NonemptyInterval.fst_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd.1 = n • s.toProd.1

theorem NonemptyInterval.snd_pow {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd.2 = s.toProd.2 ^ n

theorem NonemptyInterval.snd_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd.2 = n • s.toProd.2

@[simp]

theorem NonemptyInterval.pure_pow {α : Type u_2} [Monoid α] [Preorder α] [MulLeftMono α] [MulRightMono α] (a : α) (n : ℕ) : NonemptyInterval.pure a ^ n = NonemptyInterval.pure (a ^ n)

@[simp]

theorem NonemptyInterval.pure_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [AddLeftMono α] [AddRightMono α] (a : α) (n : ℕ) : n • NonemptyInterval.pure a = NonemptyInterval.pure (n • a)

instance NonemptyInterval.commMonoid {α : Type u_2} [OrderedCommMonoid α] : CommMonoid (NonemptyInterval α)

Equations

• NonemptyInterval.commMonoid = Function.Injective.commMonoid NonemptyInterval.toProd ⋯ ⋯ ⋯ ⋯

instance NonemptyInterval.addCommMonoid {α : Type u_2} [OrderedAddCommMonoid α] : AddCommMonoid (NonemptyInterval α)

Equations

• NonemptyInterval.addCommMonoid = Function.Injective.addCommMonoid NonemptyInterval.toProd ⋯ ⋯ ⋯ ⋯

instance Interval.mulOneClass {α : Type u_2} [OrderedCommMonoid α] : MulOneClass (Interval α)

Equations

• Interval.mulOneClass = MulOneClass.mk ⋯ ⋯

instance Interval.addZeroClass {α : Type u_2} [OrderedAddCommMonoid α] : AddZeroClass (Interval α)

Equations

• Interval.addZeroClass = AddZeroClass.mk ⋯ ⋯

instance Interval.commMonoid {α : Type u_2} [OrderedCommMonoid α] : CommMonoid (Interval α)

Equations

• Interval.commMonoid = CommMonoid.mk ⋯

instance Interval.addCommMonoid {α : Type u_2} [OrderedAddCommMonoid α] : AddCommMonoid (Interval α)

Equations

• Interval.addCommMonoid = AddCommMonoid.mk ⋯

@[simp]

theorem NonemptyInterval.coe_pow_interval {α : Type u_2} [OrderedCommMonoid α] (s : NonemptyInterval α) (n : ℕ) : ↑(s ^ n) = ↑s ^ n

theorem NonemptyInterval.coe_nsmul_interval {α : Type u_2} [OrderedAddCommMonoid α] (s : NonemptyInterval α) (n : ℕ) : ↑(n • s) = n • ↑s

theorem Interval.bot_pow {α : Type u_2} [OrderedCommMonoid α] {n : ℕ} : n ≠ 0 → ⊥ ^ n = ⊥

theorem Interval.bot_nsmul {α : Type u_2} [OrderedAddCommMonoid α] {n : ℕ} : n ≠ 0 → n • ⊥ = ⊥

Subtraction

Subtraction is defined more generally than division so that it applies to ℕ (and OrderedDiv is not a thing and probably should not become one).

instance instSubNonemptyInterval {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] : Sub (NonemptyInterval α)

Equations

• instSubNonemptyInterval = { sub := fun (s t : NonemptyInterval α) => { toProd := (s.toProd.1 - t.toProd.2, s.toProd.2 - t.toProd.1), fst_le_snd := ⋯ } }

instance instSubInterval {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] : Sub (Interval α)

Equations

• instSubInterval = { sub := Option.map₂ Sub.sub }

@[simp]

theorem NonemptyInterval.fst_sub {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s t : NonemptyInterval α) : (s - t).toProd.1 = s.toProd.1 - t.toProd.2

@[simp]

theorem NonemptyInterval.snd_sub {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s t : NonemptyInterval α) : (s - t).toProd.2 = s.toProd.2 - t.toProd.1

@[simp]

theorem NonemptyInterval.coe_sub_interval {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s t : NonemptyInterval α) : ↑(s - t) = ↑s - ↑t

theorem NonemptyInterval.sub_mem_sub {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s t : NonemptyInterval α) {a b : α} (ha : a ∈ s) (hb : b ∈ t) : a - b ∈ s - t

@[simp]

theorem NonemptyInterval.pure_sub_pure {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (a b : α) : NonemptyInterval.pure a - NonemptyInterval.pure b = NonemptyInterval.pure (a - b)

@[simp]

theorem Interval.bot_sub {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (t : Interval α) : ⊥ - t = ⊥

@[simp]

theorem Interval.sub_bot {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [AddLeftMono α] (s : Interval α) : s - ⊥ = ⊥

Division in ordered groups

Note that this division does not apply to ℚ or ℝ.

instance instDivNonemptyInterval {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] : Div (NonemptyInterval α)

Equations

• instDivNonemptyInterval = { div := fun (s t : NonemptyInterval α) => { toProd := (s.toProd.1 / t.toProd.2, s.toProd.2 / t.toProd.1), fst_le_snd := ⋯ } }

instance instDivInterval {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] : Div (Interval α)

Equations

• instDivInterval = { div := Option.map₂ fun (x1 x2 : NonemptyInterval α) => x1 / x2 }

@[simp]

theorem NonemptyInterval.fst_div {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s t : NonemptyInterval α) : (s / t).toProd.1 = s.toProd.1 / t.toProd.2

@[simp]

theorem NonemptyInterval.snd_div {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s t : NonemptyInterval α) : (s / t).toProd.2 = s.toProd.2 / t.toProd.1

@[simp]

theorem NonemptyInterval.coe_div_interval {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s t : NonemptyInterval α) : ↑(s / t) = ↑s / ↑t

theorem NonemptyInterval.div_mem_div {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s t : NonemptyInterval α) (a b : α) (ha : a ∈ s) (hb : b ∈ t) : a / b ∈ s / t

@[simp]

theorem NonemptyInterval.pure_div_pure {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (a b : α) : NonemptyInterval.pure a / NonemptyInterval.pure b = NonemptyInterval.pure (a / b)

@[simp]

theorem Interval.bot_div {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (t : Interval α) : ⊥ / t = ⊥

@[simp]

theorem Interval.div_bot {α : Type u_2} [Preorder α] [CommGroup α] [MulLeftMono α] (s : Interval α) : s / ⊥ = ⊥

Negation/inversion

instance instInvNonemptyInterval {α : Type u_2} [OrderedCommGroup α] : Inv (NonemptyInterval α)

Equations

• instInvNonemptyInterval = { inv := fun (s : NonemptyInterval α) => { toProd := (s.toProd.2⁻¹, s.toProd.1⁻¹), fst_le_snd := ⋯ } }

instance instNegNonemptyInterval {α : Type u_2} [OrderedAddCommGroup α] : Neg (NonemptyInterval α)

Equations

• instNegNonemptyInterval = { neg := fun (s : NonemptyInterval α) => { toProd := (-s.toProd.2, -s.toProd.1), fst_le_snd := ⋯ } }

instance instInvInterval {α : Type u_2} [OrderedCommGroup α] : Inv (Interval α)

Equations

• instInvInterval = { inv := Option.map Inv.inv }

instance instNegInterval {α : Type u_2} [OrderedAddCommGroup α] : Neg (Interval α)

Equations

• instNegInterval = { neg := Option.map Neg.neg }

@[simp]

theorem NonemptyInterval.fst_inv {α : Type u_2} [OrderedCommGroup α] (s : NonemptyInterval α) : s⁻¹.toProd.1 = s.toProd.2⁻¹

@[simp]

theorem NonemptyInterval.fst_neg {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : (-s).toProd.1 = -s.toProd.2

@[simp]

theorem NonemptyInterval.snd_inv {α : Type u_2} [OrderedCommGroup α] (s : NonemptyInterval α) : s⁻¹.toProd.2 = s.toProd.1⁻¹

@[simp]

theorem NonemptyInterval.snd_neg {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : (-s).toProd.2 = -s.toProd.1

@[simp]

theorem NonemptyInterval.coe_inv_interval {α : Type u_2} [OrderedCommGroup α] (s : NonemptyInterval α) : ↑s⁻¹ = (↑s)⁻¹

@[simp]

theorem NonemptyInterval.coe_neg_interval {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : ↑(-s) = -↑s

theorem NonemptyInterval.inv_mem_inv {α : Type u_2} [OrderedCommGroup α] (s : NonemptyInterval α) (a : α) (ha : a ∈ s) : a⁻¹ ∈ s⁻¹

theorem NonemptyInterval.neg_mem_neg {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) (a : α) (ha : a ∈ s) : -a ∈ -s

@[simp]

theorem NonemptyInterval.inv_pure {α : Type u_2} [OrderedCommGroup α] (a : α) : (NonemptyInterval.pure a)⁻¹ = NonemptyInterval.pure a⁻¹

@[simp]

theorem NonemptyInterval.neg_pure {α : Type u_2} [OrderedAddCommGroup α] (a : α) : -NonemptyInterval.pure a = NonemptyInterval.pure (-a)

@[simp]

theorem Interval.inv_bot {α : Type u_2} [OrderedCommGroup α] : ⊥⁻¹ = ⊥

@[simp]

theorem Interval.neg_bot {α : Type u_2} [OrderedAddCommGroup α] : -⊥ = ⊥

theorem NonemptyInterval.mul_eq_one_iff {α : Type u_2} [OrderedCommGroup α] {s t : NonemptyInterval α} : s * t = 1 ↔ ∃ (a : α) (b : α), s = NonemptyInterval.pure a ∧ t = NonemptyInterval.pure b ∧ a * b = 1

theorem NonemptyInterval.add_eq_zero_iff {α : Type u_2} [OrderedAddCommGroup α] {s t : NonemptyInterval α} : s + t = 0 ↔ ∃ (a : α) (b : α), s = NonemptyInterval.pure a ∧ t = NonemptyInterval.pure b ∧ a + b = 0

instance NonemptyInterval.subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] : SubtractionCommMonoid (NonemptyInterval α)

Equations

• NonemptyInterval.subtractionCommMonoid = SubtractionCommMonoid.mk ⋯

instance NonemptyInterval.divisionCommMonoid {α : Type u_2} [OrderedCommGroup α] : DivisionCommMonoid (NonemptyInterval α)

Equations

• NonemptyInterval.divisionCommMonoid = DivisionCommMonoid.mk ⋯

theorem Interval.mul_eq_one_iff {α : Type u_2} [OrderedCommGroup α] {s t : Interval α} : s * t = 1 ↔ ∃ (a : α) (b : α), s = Interval.pure a ∧ t = Interval.pure b ∧ a * b = 1

theorem Interval.add_eq_zero_iff {α : Type u_2} [OrderedAddCommGroup α] {s t : Interval α} : s + t = 0 ↔ ∃ (a : α) (b : α), s = Interval.pure a ∧ t = Interval.pure b ∧ a + b = 0

instance Interval.subtractionCommMonoid {α : Type u} [OrderedAddCommGroup α] : SubtractionCommMonoid (Interval α)

Equations

• Interval.subtractionCommMonoid = SubtractionCommMonoid.mk ⋯

instance Interval.divisionCommMonoid {α : Type u_2} [OrderedCommGroup α] : DivisionCommMonoid (Interval α)

Equations

• Interval.divisionCommMonoid = DivisionCommMonoid.mk ⋯

def NonemptyInterval.length {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : α

The length of an interval is its first component minus its second component. This measures the accuracy of the approximation by an interval.

Equations

• s.length = s.toProd.2 - s.toProd.1

@[simp]

theorem NonemptyInterval.length_nonneg {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : 0 ≤ s.length

@[simp]

theorem NonemptyInterval.length_pure {α : Type u_2} [OrderedAddCommGroup α] (a : α) : (NonemptyInterval.pure a).length = 0

@[simp]

theorem NonemptyInterval.length_zero {α : Type u_2} [OrderedAddCommGroup α] : NonemptyInterval.length 0 = 0

@[simp]

theorem NonemptyInterval.length_neg {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) : (-s).length = s.length

@[simp]

theorem NonemptyInterval.length_add {α : Type u_2} [OrderedAddCommGroup α] (s t : NonemptyInterval α) : (s + t).length = s.length + t.length

@[simp]

theorem NonemptyInterval.length_sub {α : Type u_2} [OrderedAddCommGroup α] (s t : NonemptyInterval α) : (s - t).length = s.length + t.length

@[simp]

theorem NonemptyInterval.length_sum {ι : Type u_1} {α : Type u_2} [OrderedAddCommGroup α] (f : ι → NonemptyInterval α) (s : Finset ι) : (∑ i ∈ s, f i).length = ∑ i ∈ s, (f i).length

def Interval.length {α : Type u_2} [OrderedAddCommGroup α] : Interval α → α

The length of an interval is its first component minus its second component. This measures the accuracy of the approximation by an interval.

Equations

• Interval.length none = 0
• Interval.length (some s) = s.length

@[simp]

theorem Interval.length_nonneg {α : Type u_2} [OrderedAddCommGroup α] (s : Interval α) : 0 ≤ s.length

@[simp]

theorem Interval.length_pure {α : Type u_2} [OrderedAddCommGroup α] (a : α) : (Interval.pure a).length = 0

@[simp]

theorem Interval.length_zero {α : Type u_2} [OrderedAddCommGroup α] : Interval.length 0 = 0

@[simp]

theorem Interval.length_neg {α : Type u_2} [OrderedAddCommGroup α] (s : Interval α) : (-s).length = s.length

theorem Interval.length_add_le {α : Type u_2} [OrderedAddCommGroup α] (s t : Interval α) : (s + t).length ≤ s.length + t.length

theorem Interval.length_sub_le {α : Type u_2} [OrderedAddCommGroup α] (s t : Interval α) : (s - t).length ≤ s.length + t.length

theorem Interval.length_sum_le {ι : Type u_1} {α : Type u_2} [OrderedAddCommGroup α] (f : ι → Interval α) (s : Finset ι) : (∑ i ∈ s, f i).length ≤ ∑ i ∈ s, (f i).length

def Mathlib.Meta.Positivity.evalNonemptyIntervalLength : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: The length of an interval is always nonnegative.

def Mathlib.Meta.Positivity.evalIntervalLength : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: The length of an interval is always nonnegative.