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

Equations

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

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

Equations

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

@[simp]

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

@[simp]

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

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

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

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

@[simp]

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

@[simp]

theorem NonemptyInterval.coe_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

theorem NonemptyInterval.one_mem_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

@[simp]

theorem Interval.coe_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

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

Addition/multiplication

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

theorem instAddNonemptyInterval.proof_1 {α : Type u_1} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : s.toProd.1 + t.toProd.1 ≤ s.toProd.2 + t.toProd.2

instance instAddNonemptyInterval {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : Add (NonemptyInterval α)

Equations

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

instance instMulNonemptyInterval {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : Mul (NonemptyInterval α)

Equations

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

instance instAddInterval {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : Add (Interval α)

Equations

• instAddInterval = { add := Option.map₂ fun (x x_1 : NonemptyInterval α) => x + x_1 }

instance instMulInterval {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : Mul (Interval α)

Equations

• instMulInterval = { mul := Option.map₂ fun (x x_1 : NonemptyInterval α) => x * x_1 }

@[simp]

theorem NonemptyInterval.toProd_add {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s + t).toProd = s.toProd + t.toProd

@[simp]

theorem NonemptyInterval.toProd_mul {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s * t).toProd = s.toProd * t.toProd

theorem NonemptyInterval.fst_add {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s + t).toProd.1 = s.toProd.1 + t.toProd.1

theorem NonemptyInterval.fst_mul {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s * t).toProd.1 = s.toProd.1 * t.toProd.1

theorem NonemptyInterval.snd_add {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s + t).toProd.2 = s.toProd.2 + t.toProd.2

theorem NonemptyInterval.snd_mul {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s * t).toProd.2 = s.toProd.2 * t.toProd.2

@[simp]

theorem NonemptyInterval.coe_add_interval {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : ↑(s + t) = ↑s + ↑t

@[simp]

theorem NonemptyInterval.coe_mul_interval {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : ↑(s * t) = ↑s * ↑t

@[simp]

theorem NonemptyInterval.pure_add_pure {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (b : α) : NonemptyInterval.pure a + NonemptyInterval.pure b = NonemptyInterval.pure (a + b)

@[simp]

theorem NonemptyInterval.pure_mul_pure {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (b : α) : NonemptyInterval.pure a * NonemptyInterval.pure b = NonemptyInterval.pure (a * b)

@[simp]

theorem Interval.bot_add {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (t : Interval α) : ⊥ + t = ⊥

@[simp]

theorem Interval.bot_mul {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (t : Interval α) : ⊥ * t = ⊥

theorem Interval.add_bot {α : Type u_2} [Preorder α] [Add α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : Interval α) : s + ⊥ = ⊥

@[simp]

theorem Interval.mul_bot {α : Type u_2} [Preorder α] [Mul α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Interval α) : s * ⊥ = ⊥

Powers

instance NonemptyInterval.hasNSMul {α : Type u_2} [AddMonoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : 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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : 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_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd = n • s.toProd

@[simp]

theorem NonemptyInterval.toProd_pow {α : Type u_2} [Monoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd = s.toProd ^ n

theorem NonemptyInterval.fst_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd.1 = n • s.toProd.1

theorem NonemptyInterval.fst_pow {α : Type u_2} [Monoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd.1 = s.toProd.1 ^ n

theorem NonemptyInterval.snd_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd.2 = n • s.toProd.2

theorem NonemptyInterval.snd_pow {α : Type u_2} [Monoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (n : ℕ) : (s ^ n).toProd.2 = s.toProd.2 ^ n

@[simp]

theorem NonemptyInterval.pure_nsmul {α : Type u_2} [AddMonoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (n : ℕ) : n • NonemptyInterval.pure a = NonemptyInterval.pure (n • a)

@[simp]

theorem NonemptyInterval.pure_pow {α : Type u_2} [Monoid α] [Preorder α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] [CovariantClass α α (Function.swap fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (n : ℕ) : NonemptyInterval.pure a ^ n = NonemptyInterval.pure (a ^ n)

theorem NonemptyInterval.addCommMonoid.proof_4 {α : Type u_1} [OrderedAddCommMonoid α] (s : NonemptyInterval α) (n : ℕ) : (n • s).toProd = n • s.toProd

theorem NonemptyInterval.addCommMonoid.proof_2 {α : Type u_1} [OrderedAddCommMonoid α] : NonemptyInterval.toProd 0 = 0

theorem NonemptyInterval.addCommMonoid.proof_3 {α : Type u_1} [OrderedAddCommMonoid α] (s : NonemptyInterval α) (t : NonemptyInterval α) : (s + t).toProd = s.toProd + t.toProd

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

Equations

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

theorem NonemptyInterval.addCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommMonoid α] : Function.Injective NonemptyInterval.toProd

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

Equations

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

theorem Interval.addZeroClass.proof_2 {α : Type u_1} [OrderedAddCommMonoid α] (s : Interval α) : Option.map₂ (fun (x x_1 : NonemptyInterval α) => x + x_1) s (some 0) = s

theorem Interval.addZeroClass.proof_1 {α : Type u_1} [OrderedAddCommMonoid α] (s : Interval α) : Option.map₂ (fun (x x_1 : NonemptyInterval α) => x + x_1) (some 0) s = s

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

Equations

• Interval.addZeroClass = AddZeroClass.mk ⋯ ⋯

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

Equations

• Interval.mulOneClass = MulOneClass.mk ⋯ ⋯

theorem Interval.addCommMonoid.proof_4 {α : Type u_1} [OrderedAddCommMonoid α] : ∀ (x : Interval α), nsmulRec 0 x = nsmulRec 0 x

theorem Interval.addCommMonoid.proof_1 {α : Type u_1} [OrderedAddCommMonoid α] : ∀ (x x_1 x_2 : Interval α), Option.map₂ (fun (x x_3 : NonemptyInterval α) => x + x_3) (Option.map₂ (fun (x x_3 : NonemptyInterval α) => x + x_3) x x_1) x_2 = Option.map₂ (fun (x x_3 : NonemptyInterval α) => x + x_3) x (Option.map₂ (fun (x x_3 : NonemptyInterval α) => x + x_3) x_1 x_2)

theorem Interval.addCommMonoid.proof_6 {α : Type u_1} [OrderedAddCommMonoid α] : ∀ (x x_1 : Interval α), Option.map₂ (fun (x x_2 : NonemptyInterval α) => x + x_2) x x_1 = Option.map₂ (fun (x x_2 : NonemptyInterval α) => x + x_2) x_1 x

theorem Interval.addCommMonoid.proof_5 {α : Type u_1} [OrderedAddCommMonoid α] : ∀ (n : ℕ) (x : Interval α), nsmulRec (n + 1) x = nsmulRec (n + 1) x

theorem Interval.addCommMonoid.proof_2 {α : Type u_1} [OrderedAddCommMonoid α] (a : Interval α) : 0 + a = a

theorem Interval.addCommMonoid.proof_3 {α : Type u_1} [OrderedAddCommMonoid α] (a : Interval α) : a + 0 = a

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

Equations

• Interval.addCommMonoid = let __src := Interval.addZeroClass; AddCommMonoid.mk ⋯

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

Equations

• Interval.commMonoid = let __src := Interval.mulOneClass; CommMonoid.mk ⋯

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

@[simp]

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

abbrev Interval.bot_nsmul.match_1 (motive : (x : ℕ) → x ≠ 0 → Prop) : ∀ (x : ℕ) (x_1 : x ≠ 0), (∀ (h : 0 ≠ 0), motive 0 h) → (∀ (n : ℕ) (x : n.succ ≠ 0), motive n.succ x) → motive x x_1

Equations

• ⋯ = ⋯

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

theorem Interval.bot_pow {α : Type u_2} [OrderedCommMonoid 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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : 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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] : 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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : ↑(s - t) = ↑s - ↑t

theorem NonemptyInterval.sub_mem_sub {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) {a : α} {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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (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] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (t : Interval α) : ⊥ - t = ⊥

@[simp]

theorem Interval.sub_bot {α : Type u_2} [Preorder α] [AddCommSemigroup α] [Sub α] [OrderedSub α] [CovariantClass α α (fun (x x_1 : α) => x + x_1) fun (x x_1 : α) => x ≤ x_1] (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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : 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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] : Div (Interval α)

Equations

• instDivInterval = { div := Option.map₂ fun (x x_1 : NonemptyInterval α) => x / x_1 }

@[simp]

theorem NonemptyInterval.fst_div {α : Type u_2} [Preorder α] [CommGroup α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) : ↑(s / t) = ↑s / ↑t

theorem NonemptyInterval.div_mem_div {α : Type u_2} [Preorder α] [CommGroup α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : NonemptyInterval α) (t : NonemptyInterval α) (a : α) (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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (a : α) (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] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (t : Interval α) : ⊥ / t = ⊥

@[simp]

theorem Interval.div_bot {α : Type u_2} [Preorder α] [CommGroup α] [CovariantClass α α (fun (x x_1 : α) => x * x_1) fun (x x_1 : α) => x ≤ x_1] (s : Interval α) : s / ⊥ = ⊥

Negation/inversion

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 := ⋯ } }

theorem instNegNonemptyInterval.proof_1 {α : Type u_1} [OrderedAddCommGroup α] (s : NonemptyInterval α) : -s.toProd.2 ≤ -s.toProd.1

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

Equations

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

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

Equations

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

@[simp]

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

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

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

@[simp]

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

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

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

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

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

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

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

Equations

• NonemptyInterval.subtractionCommMonoid = let __src := NonemptyInterval.addCommMonoid; SubtractionCommMonoid.mk ⋯

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

Equations

• NonemptyInterval.divisionCommMonoid = let __src := NonemptyInterval.commMonoid; DivisionCommMonoid.mk ⋯

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

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

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

Equations

• Interval.subtractionCommMonoid = let __src := Interval.addCommMonoid; SubtractionCommMonoid.mk ⋯

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

Equations

• Interval.divisionCommMonoid = let __src := Interval.commMonoid; 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 : NonemptyInterval α) (t : NonemptyInterval α) : (s + t).length = s.length + t.length

@[simp]

theorem NonemptyInterval.length_sub {α : Type u_2} [OrderedAddCommGroup α] (s : NonemptyInterval α) (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) : (s.sum fun (i : ι) => f i).length = s.sum fun (i : ι) => (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

• x.length = match x with | none => 0 | 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 : Interval α) (t : Interval α) : (s + t).length ≤ s.length + t.length

theorem Interval.length_sub_le {α : Type u_2} [OrderedAddCommGroup α] (s : Interval α) (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) : (s.sum fun (i : ι) => f i).length ≤ s.sum fun (i : ι) => (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.