Documentation

Mathlib.Data.Real.Sqrt

Square root of a real number #

In this file we define

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 :

Square root of a nonnegative real number.

Equations
    Instances For
      @[simp]
      theorem NNReal.sq_sqrt (x : NNReal) :
      sqrt x ^ 2 = x
      @[simp]
      theorem NNReal.sqrt_sq (x : NNReal) :
      sqrt (x ^ 2) = x
      @[simp]
      theorem NNReal.mul_self_sqrt (x : NNReal) :
      sqrt x * sqrt x = x
      @[simp]
      theorem NNReal.sqrt_mul_self (x : NNReal) :
      sqrt (x * x) = x
      theorem NNReal.sqrt_lt_sqrt {x y : NNReal} :
      sqrt x < sqrt y x < y
      theorem NNReal.sqrt_eq_iff_eq_sq {x y : NNReal} :
      sqrt x = y x = y ^ 2
      @[simp]
      theorem NNReal.sqrt_eq_zero {x : NNReal} :
      sqrt x = 0 x = 0
      @[simp]
      theorem NNReal.sqrt_eq_one {x : NNReal} :
      sqrt x = 1 x = 1
      @[simp]
      theorem NNReal.sqrt_zero :
      sqrt 0 = 0
      @[simp]
      theorem NNReal.sqrt_one :
      sqrt 1 = 1
      @[simp]
      theorem NNReal.sqrt_le_one {x : NNReal} :
      sqrt x 1 x 1
      @[simp]
      theorem NNReal.one_le_sqrt {x : NNReal} :
      1 sqrt x 1 x
      theorem NNReal.sqrt_mul (x y : NNReal) :
      sqrt (x * y) = sqrt x * sqrt y
      noncomputable def NNReal.sqrtHom :

      NNReal.sqrt as a MonoidWithZeroHom.

      Equations
        Instances For
          theorem NNReal.sqrt_div (x y : NNReal) :
          sqrt (x / y) = sqrt x / sqrt y
          @[simp]
          theorem NNReal.sqrt_pos {x : NNReal} :
          0 < sqrt x 0 < x
          theorem NNReal.sqrt_pos_of_pos {x : NNReal} :
          0 < x0 < sqrt x

          Alias of the reverse direction of NNReal.sqrt_pos.

          noncomputable def Real.sqrt (x : ) :

          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
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              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
                Instances For
                  @[simp]
                  theorem Real.coe_sqrt {x : NNReal} :
                  (NNReal.sqrt x) = x
                  theorem Real.sqrt_eq_zero_of_nonpos {x : } (h : x 0) :
                  x = 0
                  @[simp]
                  theorem Real.sqrt_nonneg (x : ) :
                  0 x
                  @[simp]
                  theorem Real.mul_self_sqrt {x : } (h : 0 x) :
                  x * x = x
                  @[simp]
                  theorem Real.sqrt_mul_self {x : } (h : 0 x) :
                  (x * x) = x
                  theorem Real.sqrt_eq_cases {x y : } :
                  x = y y * y = x 0 y x < 0 y = 0
                  theorem Real.sqrt_eq_iff_mul_self_eq {x y : } (hx : 0 x) (hy : 0 y) :
                  x = y x = y * y
                  theorem Real.sqrt_eq_iff_mul_self_eq_of_pos {x y : } (h : 0 < y) :
                  x = y y * y = x
                  @[simp]
                  theorem Real.sqrt_eq_one {x : } :
                  x = 1 x = 1
                  @[simp]
                  theorem Real.sq_sqrt {x : } (h : 0 x) :
                  x ^ 2 = x
                  @[simp]
                  theorem Real.sqrt_sq {x : } (h : 0 x) :
                  (x ^ 2) = x
                  theorem Real.sqrt_eq_iff_eq_sq {x y : } (hx : 0 x) (hy : 0 y) :
                  x = y x = y ^ 2
                  theorem Real.sqrt_sq_eq_abs (x : ) :
                  (x ^ 2) = |x|
                  @[simp]
                  theorem Real.sqrt_zero :
                  0 = 0
                  @[simp]
                  theorem Real.sqrt_one :
                  1 = 1
                  @[simp]
                  theorem Real.sqrt_le_sqrt_iff {x y : } (hy : 0 y) :
                  x y x y
                  @[simp]
                  theorem Real.sqrt_lt_sqrt_iff {x y : } (hx : 0 x) :
                  x < y x < y
                  theorem Real.sqrt_lt_sqrt_iff_of_pos {x y : } (hy : 0 < y) :
                  x < y x < y
                  theorem Real.sqrt_le_sqrt {x y : } (h : x y) :
                  theorem Real.sqrt_lt_sqrt {x y : } (hx : 0 x) (h : x < y) :
                  theorem Real.sqrt_le_left {x y : } (hy : 0 y) :
                  x y x y ^ 2
                  theorem Real.sqrt_le_iff {x y : } :
                  x y 0 y x y ^ 2
                  theorem Real.sqrt_lt {x y : } (hx : 0 x) (hy : 0 y) :
                  x < y x < y ^ 2
                  theorem Real.sqrt_lt' {x y : } (hy : 0 < y) :
                  x < y x < y ^ 2
                  theorem Real.le_sqrt {x y : } (hx : 0 x) (hy : 0 y) :
                  x y 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 y : } (hx : 0 < x) :
                  x y x ^ 2 y
                  theorem Real.abs_le_sqrt {x y : } (h : x ^ 2 y) :
                  theorem Real.sq_le {x y : } (h : 0 y) :
                  x ^ 2 y -y x x y
                  theorem Real.neg_sqrt_le_of_sq_le {x y : } (h : x ^ 2 y) :
                  theorem Real.le_sqrt_of_sq_le {x y : } (h : x ^ 2 y) :
                  x y
                  @[simp]
                  theorem Real.sqrt_inj {x y : } (hx : 0 x) (hy : 0 y) :
                  x = y x = y
                  @[simp]
                  theorem Real.sqrt_eq_zero {x : } (h : 0 x) :
                  x = 0 x = 0
                  theorem Real.sqrt_eq_zero' {x : } :
                  x = 0 x 0
                  theorem Real.sqrt_ne_zero {x : } (h : 0 x) :
                  x 0 x 0
                  theorem Real.sqrt_ne_zero' {x : } :
                  x 0 0 < x
                  @[simp]
                  theorem Real.sqrt_pos {x : } :
                  0 < x 0 < x
                  theorem Real.sqrt_pos_of_pos {x : } :
                  0 < x0 < x

                  Alias of the reverse direction of Real.sqrt_pos.

                  theorem Real.sqrt_le_sqrt_iff' {x y : } (hx : 0 < x) :
                  x y x y
                  @[simp]
                  theorem Real.one_le_sqrt {x : } :
                  1 x 1 x
                  @[simp]
                  theorem Real.sqrt_le_one {x : } :
                  x 1 x 1

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

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

                      Equations
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                          @[simp]
                          theorem Real.sqrt_mul {x : } (hx : 0 x) (y : ) :
                          (x * y) = x * y
                          @[simp]
                          theorem Real.sqrt_mul' (x : ) {y : } (hy : 0 y) :
                          (x * y) = x * y
                          @[simp]
                          theorem Real.sqrt_inv (x : ) :
                          @[simp]
                          theorem Real.sqrt_div {x : } (hx : 0 x) (y : ) :
                          (x / y) = x / y
                          @[simp]
                          theorem Real.sqrt_div' (x : ) {y : } (hy : 0 y) :
                          (x / y) = x / y
                          @[simp]
                          theorem Real.div_sqrt {x : } :
                          x / x = x
                          theorem Real.sqrt_div_self' {x : } :
                          x / x = 1 / x
                          theorem Real.lt_sqrt {x y : } (hx : 0 x) :
                          x < y x ^ 2 < y
                          theorem Real.sq_lt {x y : } :
                          x ^ 2 < y -y < x x < y
                          theorem Real.neg_sqrt_lt_of_sq_lt {x y : } (h : x ^ 2 < y) :
                          -y < x
                          theorem Real.lt_sqrt_of_sq_lt {x y : } (h : x ^ 2 < y) :
                          x < y
                          theorem Real.lt_sq_of_sqrt_lt {x y : } (h : x < y) :
                          x < y ^ 2
                          theorem Real.nat_sqrt_le_real_sqrt {a : } :
                          a.sqrt a

                          The natural square root is at most the real square root

                          theorem Real.real_sqrt_lt_nat_sqrt_succ {a : } :
                          a < a.sqrt + 1

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

                          theorem Real.real_sqrt_le_nat_sqrt_succ {a : } :
                          a a.sqrt + 1

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

                          @[simp]

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

                          @[simp]

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

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

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

                          theorem Filter.Tendsto.sqrt {α : Type u_1} {f : α} {l : Filter α} {x : } (h : Tendsto f l (nhds x)) :
                          Tendsto (fun (x : α) => (f x)) l (nhds x)
                          theorem ContinuousWithinAt.sqrt {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} {x : α} (h : ContinuousWithinAt f s x) :
                          ContinuousWithinAt (fun (x : α) => (f x)) s x
                          theorem ContinuousAt.sqrt {α : Type u_1} [TopologicalSpace α] {f : α} {x : α} (h : ContinuousAt f x) :
                          ContinuousAt (fun (x : α) => (f x)) x
                          theorem ContinuousOn.sqrt {α : Type u_1} [TopologicalSpace α] {f : α} {s : Set α} (h : ContinuousOn f s) :
                          ContinuousOn (fun (x : α) => (f x)) s
                          theorem Continuous.sqrt {α : Type u_1} [TopologicalSpace α] {f : α} (h : Continuous f) :
                          Continuous fun (x : α) => (f x)
                          theorem NNReal.sum_mul_le_sqrt_mul_sqrt {ι : Type u_2} (s : Finset ι) (f g : ιNNReal) :
                          is, f i * g i sqrt (∑ is, f i ^ 2) * sqrt (∑ is, 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 g : ιNNReal) :
                          is, sqrt (f i) * sqrt (g i) sqrt (∑ is, f i) * sqrt (∑ is, 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 g : ι) :
                          is, f i * g i (∑ is, f i ^ 2) * (∑ is, g i ^ 2)

                          Cauchy-Schwarz inequality for finsets using square roots in .

                          theorem Real.sum_sqrt_mul_sqrt_le {ι : Type u_2} {f g : ι} (s : Finset ι) (hf : ∀ (i : ι), 0 f i) (hg : ∀ (i : ι), 0 g i) :
                          is, (f i) * (g i) (∑ is, f i) * (∑ is, g i)

                          Cauchy-Schwarz inequality for finsets using square roots in .