Square root of a real number

In this file we define

• NNReal.sqrt to be the square root of a nonnegative real number.
• Real.sqrt to be the square root of a real number, defined to be zero on negative numbers.

Then we prove some basic properties of these functions.

Implementation notes

We define NNReal.sqrt as the noncomputable inverse to the function x ↦ x * x. We use general theory of inverses of strictly monotone functions to prove that NNReal.sqrt x exists. As a side effect, NNReal.sqrt is a bundled OrderIso, so for NNReal numbers we get continuity as well as theorems like NNReal.sqrt x ≤ y ↔ x ≤ y * y for free.

Then we define Real.sqrt x to be NNReal.sqrt (Real.toNNReal x).

Tags

square root

noncomputable def NNReal.sqrt : NNReal ≃o NNReal

Square root of a nonnegative real number.

Equations

• NNReal.sqrt = (NNReal.powOrderIso 2 NNReal.sqrt.proof_1).symm

@[simp]

theorem NNReal.sq_sqrt (x : NNReal) : NNReal.sqrt x ^ 2 = x

@[simp]

theorem NNReal.sqrt_sq (x : NNReal) : NNReal.sqrt (x ^ 2) = x

@[simp]

theorem NNReal.mul_self_sqrt (x : NNReal) : NNReal.sqrt x * NNReal.sqrt x = x

@[simp]

theorem NNReal.sqrt_mul_self (x : NNReal) : NNReal.sqrt (x * x) = x

theorem NNReal.sqrt_le_sqrt {x : NNReal} {y : NNReal} : NNReal.sqrt x ≤ NNReal.sqrt y ↔ x ≤ y

theorem NNReal.sqrt_lt_sqrt {x : NNReal} {y : NNReal} : NNReal.sqrt x < NNReal.sqrt y ↔ x < y

theorem NNReal.sqrt_eq_iff_eq_sq {x : NNReal} {y : NNReal} : NNReal.sqrt x = y ↔ x = y ^ 2

theorem NNReal.sqrt_le_iff_le_sq {x : NNReal} {y : NNReal} : NNReal.sqrt x ≤ y ↔ x ≤ y ^ 2

theorem NNReal.le_sqrt_iff_sq_le {x : NNReal} {y : NNReal} : x ≤ NNReal.sqrt y ↔ x ^ 2 ≤ y

@[deprecated NNReal.sqrt_le_sqrt]

theorem NNReal.sqrt_le_sqrt_iff {x : NNReal} {y : NNReal} : NNReal.sqrt x ≤ NNReal.sqrt y ↔ x ≤ y

Alias of NNReal.sqrt_le_sqrt.

@[deprecated NNReal.sqrt_lt_sqrt]

theorem NNReal.sqrt_lt_sqrt_iff {x : NNReal} {y : NNReal} : NNReal.sqrt x < NNReal.sqrt y ↔ x < y

Alias of NNReal.sqrt_lt_sqrt.

@[deprecated NNReal.sqrt_le_iff_le_sq]

theorem NNReal.sqrt_le_iff {x : NNReal} {y : NNReal} : NNReal.sqrt x ≤ y ↔ x ≤ y ^ 2

Alias of NNReal.sqrt_le_iff_le_sq.

@[deprecated NNReal.le_sqrt_iff_sq_le]

theorem NNReal.le_sqrt_iff {x : NNReal} {y : NNReal} : x ≤ NNReal.sqrt y ↔ x ^ 2 ≤ y

Alias of NNReal.le_sqrt_iff_sq_le.

@[deprecated NNReal.sqrt_eq_iff_eq_sq]

theorem NNReal.sqrt_eq_iff_sq_eq {x : NNReal} {y : NNReal} : NNReal.sqrt x = y ↔ x = y ^ 2

Alias of NNReal.sqrt_eq_iff_eq_sq.

@[simp]

theorem NNReal.sqrt_eq_zero {x : NNReal} : NNReal.sqrt x = 0 ↔ x = 0

@[simp]

theorem NNReal.sqrt_eq_one {x : NNReal} : NNReal.sqrt x = 1 ↔ x = 1

@[simp]

theorem NNReal.sqrt_zero : NNReal.sqrt 0 = 0

@[simp]

theorem NNReal.sqrt_one : NNReal.sqrt 1 = 1

@[simp]

theorem NNReal.sqrt_le_one {x : NNReal} : NNReal.sqrt x ≤ 1 ↔ x ≤ 1

@[simp]

theorem NNReal.one_le_sqrt {x : NNReal} : 1 ≤ NNReal.sqrt x ↔ 1 ≤ x

theorem NNReal.sqrt_mul (x : NNReal) (y : NNReal) : NNReal.sqrt (x * y) = NNReal.sqrt x * NNReal.sqrt y

noncomputable def NNReal.sqrtHom : NNReal →*₀ NNReal

NNReal.sqrt as a MonoidWithZeroHom.

Equations

• NNReal.sqrtHom = { toFun := ⇑NNReal.sqrt, map_zero' := NNReal.sqrt_zero, map_one' := NNReal.sqrt_one, map_mul' := NNReal.sqrt_mul }

theorem NNReal.sqrt_inv (x : NNReal) : NNReal.sqrt x⁻¹ = (NNReal.sqrt x)⁻¹

theorem NNReal.sqrt_div (x : NNReal) (y : NNReal) : NNReal.sqrt (x / y) = NNReal.sqrt x / NNReal.sqrt y

theorem NNReal.continuous_sqrt : Continuous ⇑NNReal.sqrt

@[simp]

theorem NNReal.sqrt_pos {x : NNReal} : 0 < NNReal.sqrt x ↔ 0 < x

theorem NNReal.sqrt_pos_of_pos {x : NNReal} : 0 < x → 0 < NNReal.sqrt x

Alias of the reverse direction of NNReal.sqrt_pos.

noncomputable def Real.sqrt (x : ℝ) : ℝ

The square root of a real number. This returns 0 for negative inputs.

This has notation √x. Note that √x⁻¹ is parsed as √(x⁻¹).

Equations

• x.sqrt = ↑(NNReal.sqrt x.toNNReal)

def Real.«term√_» : Lean.ParserDescr

The square root of a real number. This returns 0 for negative inputs.

This has notation √x. Note that √x⁻¹ is parsed as √(x⁻¹).

Equations

• Real.«term√_» = Lean.ParserDescr.node `Real.term√_ 1024 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol "√") (Lean.ParserDescr.cat `term 1024))

@[simp]

theorem Real.coe_sqrt {x : NNReal} : ↑(NNReal.sqrt x) = (↑x).sqrt

theorem Real.continuous_sqrt : Continuous fun (x : ℝ) => x.sqrt

theorem Real.sqrt_eq_zero_of_nonpos {x : ℝ} (h : x ≤ 0) : x.sqrt = 0

theorem Real.sqrt_nonneg (x : ℝ) : 0 ≤ x.sqrt

@[simp]

theorem Real.mul_self_sqrt {x : ℝ} (h : 0 ≤ x) : x.sqrt * x.sqrt = x

@[simp]

theorem Real.sqrt_mul_self {x : ℝ} (h : 0 ≤ x) : (x * x).sqrt = x

theorem Real.sqrt_eq_cases {x : ℝ} {y : ℝ} : x.sqrt = y ↔ y * y = x ∧ 0 ≤ y ∨ x < 0 ∧ y = 0

theorem Real.sqrt_eq_iff_mul_self_eq {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x.sqrt = y ↔ y * y = x

theorem Real.sqrt_eq_iff_mul_self_eq_of_pos {x : ℝ} {y : ℝ} (h : 0 < y) : x.sqrt = y ↔ y * y = x

@[simp]

theorem Real.sqrt_eq_one {x : ℝ} : x.sqrt = 1 ↔ x = 1

@[simp]

theorem Real.sq_sqrt {x : ℝ} (h : 0 ≤ x) : x.sqrt ^ 2 = x

@[simp]

theorem Real.sqrt_sq {x : ℝ} (h : 0 ≤ x) : (x ^ 2).sqrt = x

theorem Real.sqrt_eq_iff_sq_eq {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x.sqrt = y ↔ y ^ 2 = x

theorem Real.sqrt_mul_self_eq_abs (x : ℝ) : (x * x).sqrt = |x|

theorem Real.sqrt_sq_eq_abs (x : ℝ) : (x ^ 2).sqrt = |x|

@[simp]

theorem Real.sqrt_zero : √0 = 0

@[simp]

theorem Real.sqrt_one : √1 = 1

@[simp]

theorem Real.sqrt_le_sqrt_iff {x : ℝ} {y : ℝ} (hy : 0 ≤ y) : x.sqrt ≤ y.sqrt ↔ x ≤ y

@[simp]

theorem Real.sqrt_lt_sqrt_iff {x : ℝ} {y : ℝ} (hx : 0 ≤ x) : x.sqrt < y.sqrt ↔ x < y

theorem Real.sqrt_lt_sqrt_iff_of_pos {x : ℝ} {y : ℝ} (hy : 0 < y) : x.sqrt < y.sqrt ↔ x < y

theorem Real.sqrt_le_sqrt {x : ℝ} {y : ℝ} (h : x ≤ y) : x.sqrt ≤ y.sqrt

theorem Real.sqrt_lt_sqrt {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (h : x < y) : x.sqrt < y.sqrt

theorem Real.sqrt_le_left {x : ℝ} {y : ℝ} (hy : 0 ≤ y) : x.sqrt ≤ y ↔ x ≤ y ^ 2

theorem Real.sqrt_le_iff {x : ℝ} {y : ℝ} : x.sqrt ≤ y ↔ 0 ≤ y ∧ x ≤ y ^ 2

theorem Real.sqrt_lt {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x.sqrt < y ↔ x < y ^ 2

theorem Real.sqrt_lt' {x : ℝ} {y : ℝ} (hy : 0 < y) : x.sqrt < y ↔ x < y ^ 2

theorem Real.le_sqrt {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ y.sqrt ↔ x ^ 2 ≤ y

Note: if you want to conclude x ≤ √y, then use Real.le_sqrt_of_sq_le. If you have x > 0, consider using Real.le_sqrt'

theorem Real.le_sqrt' {x : ℝ} {y : ℝ} (hx : 0 < x) : x ≤ y.sqrt ↔ x ^ 2 ≤ y

theorem Real.abs_le_sqrt {x : ℝ} {y : ℝ} (h : x ^ 2 ≤ y) : |x| ≤ y.sqrt

theorem Real.sq_le {x : ℝ} {y : ℝ} (h : 0 ≤ y) : x ^ 2 ≤ y ↔ -y.sqrt ≤ x ∧ x ≤ y.sqrt

theorem Real.neg_sqrt_le_of_sq_le {x : ℝ} {y : ℝ} (h : x ^ 2 ≤ y) : -y.sqrt ≤ x

theorem Real.le_sqrt_of_sq_le {x : ℝ} {y : ℝ} (h : x ^ 2 ≤ y) : x ≤ y.sqrt

@[simp]

theorem Real.sqrt_inj {x : ℝ} {y : ℝ} (hx : 0 ≤ x) (hy : 0 ≤ y) : x.sqrt = y.sqrt ↔ x = y

@[simp]

theorem Real.sqrt_eq_zero {x : ℝ} (h : 0 ≤ x) : x.sqrt = 0 ↔ x = 0

theorem Real.sqrt_eq_zero' {x : ℝ} : x.sqrt = 0 ↔ x ≤ 0

theorem Real.sqrt_ne_zero {x : ℝ} (h : 0 ≤ x) : x.sqrt ≠ 0 ↔ x ≠ 0

theorem Real.sqrt_ne_zero' {x : ℝ} : x.sqrt ≠ 0 ↔ 0 < x

@[simp]

theorem Real.sqrt_pos {x : ℝ} : 0 < x.sqrt ↔ 0 < x

theorem Real.sqrt_pos_of_pos {x : ℝ} : 0 < x → 0 < x.sqrt

Alias of the reverse direction of Real.sqrt_pos.

theorem Real.sqrt_le_sqrt_iff' {x : ℝ} {y : ℝ} (hx : 0 < x) : x.sqrt ≤ y.sqrt ↔ x ≤ y

@[simp]

theorem Real.one_le_sqrt {x : ℝ} : 1 ≤ x.sqrt ↔ 1 ≤ x

@[simp]

theorem Real.sqrt_le_one {x : ℝ} : x.sqrt ≤ 1 ↔ x ≤ 1

def Mathlib.Meta.Positivity.evalNNRealSqrt : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: a square root of a strictly positive nonnegative real is positive.

def Mathlib.Meta.Positivity.evalSqrt : Mathlib.Meta.Positivity.PositivityExt

Extension for the positivity tactic: a square root is nonnegative, and is strictly positive if its input is.

@[simp]

theorem Real.sqrt_mul {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : (x * y).sqrt = x.sqrt * y.sqrt

@[simp]

theorem Real.sqrt_mul' (x : ℝ) {y : ℝ} (hy : 0 ≤ y) : (x * y).sqrt = x.sqrt * y.sqrt

@[simp]

theorem Real.sqrt_inv (x : ℝ) : x⁻¹.sqrt = x.sqrt⁻¹

@[simp]

theorem Real.sqrt_div {x : ℝ} (hx : 0 ≤ x) (y : ℝ) : (x / y).sqrt = x.sqrt / y.sqrt

@[simp]

theorem Real.sqrt_div' (x : ℝ) {y : ℝ} (hy : 0 ≤ y) : (x / y).sqrt = x.sqrt / y.sqrt

@[simp]

theorem Real.div_sqrt {x : ℝ} : x / x.sqrt = x.sqrt

theorem Real.sqrt_div_self' {x : ℝ} : x.sqrt / x = 1 / x.sqrt

theorem Real.sqrt_div_self {x : ℝ} : x.sqrt / x = x.sqrt⁻¹

theorem Real.lt_sqrt {x : ℝ} {y : ℝ} (hx : 0 ≤ x) : x < y.sqrt ↔ x ^ 2 < y

theorem Real.sq_lt {x : ℝ} {y : ℝ} : x ^ 2 < y ↔ -y.sqrt < x ∧ x < y.sqrt

theorem Real.neg_sqrt_lt_of_sq_lt {x : ℝ} {y : ℝ} (h : x ^ 2 < y) : -y.sqrt < x

theorem Real.lt_sqrt_of_sq_lt {x : ℝ} {y : ℝ} (h : x ^ 2 < y) : x < y.sqrt

theorem Real.lt_sq_of_sqrt_lt {x : ℝ} {y : ℝ} (h : x.sqrt < y) : x < y ^ 2

theorem Real.nat_sqrt_le_real_sqrt {a : ℕ} : ↑a.sqrt ≤ (↑a).sqrt

The natural square root is at most the real square root

theorem Real.real_sqrt_le_nat_sqrt_succ {a : ℕ} : (↑a).sqrt ≤ ↑a.sqrt + 1

The real square root is at most the natural square root plus one

theorem Real.real_sqrt_lt_nat_sqrt_succ {a : ℕ} : (↑a).sqrt < ↑a.sqrt + 1

The real square root is less than the natural square root plus one

@[simp]

theorem Real.floor_real_sqrt_eq_nat_sqrt {a : ℕ} : ⌊(↑a).sqrt⌋ = ↑a.sqrt

The floor of the real square root is the same as the natural square root.

@[simp]

theorem Real.nat_floor_real_sqrt_eq_nat_sqrt {a : ℕ} : ⌊(↑a).sqrt⌋₊ = a.sqrt

The natural floor of the real square root is the same as the natural square root.

theorem Real.sqrt_one_add_le {x : ℝ} (h : -1 ≤ x) : (1 + x).sqrt ≤ 1 + x / 2

Bernoulli's inequality for exponent 1 / 2, stated using sqrt.

instance Real.instStarOrderedRing : StarOrderedRing ℝ

Although the instance RCLike.toStarOrderedRing exists, it is locked behind the ComplexOrder scope because currently the order on ℂ is not enabled globally. But we want StarOrderedRing ℝ to be available globally, so we include this instance separately. In addition, providing this instance here makes it available earlier in the import hierarchy; otherwise in order to access it we would need to import Mathlib.Analysis.RCLike.Basic.

Equations

• Real.instStarOrderedRing = ⋯

instance NNReal.instStarOrderedRing : StarOrderedRing NNReal

Equations

• NNReal.instStarOrderedRing = ⋯

theorem Filter.Tendsto.sqrt {α : Type u_1} {f : α → ℝ} {l : Filter α} {x : ℝ} (h : Filter.Tendsto f l (nhds x)) : Filter.Tendsto (fun (x : α) => (f x).sqrt) l (nhds x.sqrt)

theorem ContinuousWithinAt.sqrt {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) : ContinuousWithinAt (fun (x : α) => (f x).sqrt) s x

theorem ContinuousAt.sqrt {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {x : α} (h : ContinuousAt f x) : ContinuousAt (fun (x : α) => (f x).sqrt) x

theorem ContinuousOn.sqrt {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} {s : Set α} (h : ContinuousOn f s) : ContinuousOn (fun (x : α) => (f x).sqrt) s

theorem Continuous.sqrt {α : Type u_1} [TopologicalSpace α] {f : α → ℝ} (h : Continuous f) : Continuous fun (x : α) => (f x).sqrt

theorem NNReal.sum_mul_le_sqrt_mul_sqrt {ι : Type u_2} (s : Finset ι) (f : ι → NNReal) (g : ι → NNReal) : (s.sum fun (i : ι) => f i * g i) ≤ NNReal.sqrt (s.sum fun (i : ι) => f i ^ 2) * NNReal.sqrt (s.sum fun (i : ι) => g i ^ 2)

Cauchy-Schwarz inequality for finsets using square roots in ℝ≥0.

theorem NNReal.sum_sqrt_mul_sqrt_le {ι : Type u_2} (s : Finset ι) (f : ι → NNReal) (g : ι → NNReal) : (s.sum fun (i : ι) => NNReal.sqrt (f i) * NNReal.sqrt (g i)) ≤ NNReal.sqrt (s.sum fun (i : ι) => f i) * NNReal.sqrt (s.sum fun (i : ι) => g i)

Cauchy-Schwarz inequality for finsets using square roots in ℝ≥0.

theorem Real.sum_mul_le_sqrt_mul_sqrt {ι : Type u_2} (s : Finset ι) (f : ι → ℝ) (g : ι → ℝ) : (s.sum fun (i : ι) => f i * g i) ≤ (s.sum fun (i : ι) => f i ^ 2).sqrt * (s.sum fun (i : ι) => g i ^ 2).sqrt

Cauchy-Schwarz inequality for finsets using square roots in ℝ.

theorem Real.sum_sqrt_mul_sqrt_le {ι : Type u_2} {f : ι → ℝ} {g : ι → ℝ} (s : Finset ι) (hf : ∀ (i : ι), 0 ≤ f i) (hg : ∀ (i : ι), 0 ≤ g i) : (s.sum fun (i : ι) => (f i).sqrt * (g i).sqrt) ≤ (s.sum fun (i : ι) => f i).sqrt * (s.sum fun (i : ι) => g i).sqrt

Cauchy-Schwarz inequality for finsets using square roots in ℝ.