introvoyz041/TaoAnalysisBench huggingface dataset:
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.Real

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.ratPow (x:Real) (q:ℚ) : Real := (x.root q.den)^(q.num)

noncomputable instance Real.instRatPow : Pow Real ℚ where
  pow x q := x.ratPow q

namespace Chapter5

/-- Lemma 5.6.9(a) / Exercise 5.6.2 -/
theorem Real.ratPow_pos {x:Real} (hx: x > 0) (q:ℚ) : x^q > 0 := by
end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter5

/-- Lemma 5.6.9(a) / Exercise 5.6.2 -/
theorem Real.ratPow_pos {x : ℝ} (hx : x > 0) (q : ℚ) : x ^ (q : ℝ) > 0 := by
end Chapter5

end TaoBench
import Mathlib.Data.Real.Basic
import Mathlib.Data.Rat.Basic

namespace Chapter5

/-- Definition 5.6.2.  We permit ``junk values'' when `x` is negative or `n` vanishes. -/
noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

/-- Definition 5.6.7 -/
noncomputable abbrev Real.ratPow (x:Real) (q:ℚ) : Real := (Real.root x q.den)^(q.num)

noncomputable instance Real.instRatPow : Pow Real ℚ where
  pow x q := Real.ratPow x q

/-- Lemma 5.6.9(b) / Exercise 5.6.2 -/
theorem Real.ratPow_ratPow {x:Real} (hx: x > 0) (q r:ℚ) : (x^q)^r = x^(q*r) := by
end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.ratPow_ratPow {x : ℝ} (hx : x > 0) (q r : ℚ) :
    (x ^ (q : ℝ)) ^ (r : ℝ) = x ^ ((q * r : ℚ) : ℝ) := by
end TaoBench
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.rat_between {x y : Real} (hxy : x < y) :
    ∃ q : ℚ, x < (q : Real) ∧ (q : Real) < y := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic

namespace Chapter5

abbrev Real := ℝ

noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real := sSup {y : Real | y ≥ 0 ∧ y ^ n ≤ x}

/-- Lemma 5.6.6 (d) / Exercise 5.6.1 -/
theorem Real.root_mono {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  x > y ↔ x.root n > y.root n := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by
  sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by
  sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by
  sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by
  sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by
  sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by
  sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by
  sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by
  sorry

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := Real.root x 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry

theorem Real.pow_of_root {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (Real.root x n)^n = x := by
  sorry

theorem Real.root_of_pow {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  Real.root (x^n) n = x := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_mono {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : x > y ↔ Real.root x n > Real.root y n := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 5.6.6 (d) / Exercise 5.6.1, Mathlib version. -/
theorem Real.root_mono {x y : ℝ} (hx : x ≥ 0) (hy : y ≥ 0) {n : ℕ} (hn : n ≥ 1) :
  x > y ↔ x ^ (1 / (n : ℝ)) > y ^ (1 / (n : ℝ)) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Topology.Algebra.Order.LiminfLimsup

namespace Chapter5

abbrev Real := ℝ

noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real :=
  sSup {y : Real | y ≥ 0 ∧ y ^ n ≤ x}

/-- Lemma 5.6.6 (e) / Exercise 5.6.1 -/
theorem Real.root_mono_of_gt_one {x : Real} (hx: x > 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) :
  x.root k < x.root l := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
abbrev Real := ℝ

lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by sorry

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := x.root 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry

theorem Real.pow_of_root {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x.root n)^n = x := by sorry

theorem Real.root_of_pow {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x^n).root n = x := by sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_mono_of_gt_one {x : Real} (hx: x > 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) : x.root k < x.root l := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.root_mono_of_gt_one {x : ℝ} (hx : x > 1) {k l : ℕ} (hkl : k > l) (hl : l ≥ 1) :
  x ^ (1 / (k : ℝ)) < x ^ (1 / (l : ℝ)) := by
end TaoBench
import Mathlib.Data.Real.Basic

namespace Chapter5

abbrev Real := _root_.Real

noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real :=
  sSup { y : Real | y ≥ 0 ∧ y ^ n ≤ x }

/-- Lemma 5.6.6 (e) / Exercise 5.6.1 -/
theorem Real.root_mono_of_lt_one {x : Real} (hx0: 0 < x) (hx: x < 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) : x.root k > x.root l := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
abbrev Real := _root_.Real

lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by
  sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by
  sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by
  sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by
  sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by
  sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by
  sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by
  sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by
  sorry

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := x.root 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry

theorem Real.pow_of_root {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x.root n)^n = x := by
  sorry

theorem Real.root_of_pow {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x^n).root n = x := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_mono_of_lt_one {x : Real} (hx0: 0 < x) (hx: x < 1) {k l: ℕ} (hkl: k > l) (hl: l ≥ 1) : x.root k > x.root l := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

/-- Lemma 5.6.6 (e) / Exercise 5.6.1 (monotonicity of real roots for bases in (0,1)) -/
theorem Real.root_mono_of_lt_one
    {x : ℝ} (hx0 : 0 < x) (hx : x < 1) {k l : ℕ}
    (hkl : k > l) (hl : l ≥ 1) :
    x ^ (1 / (k : ℝ)) > x ^ (1 / (l : ℝ)) := by
end TaoBench
import Mathlib.Tactic
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Instances.Real

noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real := sSup { y : Real | y ≥ 0 ∧ y ^ n ≤ x }

namespace Chapter5

/-- Lemma 5.6.6 (f) / Exercise 5.6.1 -/
 theorem Real.root_mul {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : (x*y).root n = (x.root n) * (y.root n) := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by
  sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by
  sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by
  sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by
  sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by
  sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by
  sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by
  sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by
  sorry

noncomputable abbrev _root_.Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := x.root 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry

theorem Real.pow_of_root {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x.root n)^n = x := by
  sorry

theorem Real.root_of_pow {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) :
  (x^n).root n = x := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_mul {x y:Real} (hx: x ≥ 0) (hy: y ≥ 0) {n:ℕ} (hn: n ≥ 1) : (x*y).root n = (x.root n) * (y.root n) := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.root_mul {x y : Real} (hx : x ≥ 0) (hy : y ≥ 0) {n : ℕ} (hn : n ≥ 1) :
  (x * y) ^ (1 / (n : ℝ)) = x ^ (1 / (n : ℝ)) * y ^ (1 / (n : ℝ)) := by
end TaoBench
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Algebra.Order

noncomputable section

namespace Chapter5

/-- Definition 5.6.2.  We permit ``junk values'' when `x` is negative or `n` vanishes. -/
noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

end Chapter5

/-- Alias to enable dot-notation `x.root n` with the ambient `Real` type. -/
noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real := Chapter5.Real.root x n

namespace Chapter5

/-- Lemma 5.6.6 (c) / Exercise 5.6.1 -/
theorem Real.root_nonneg {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n ≥ 0 := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
open Chapter5

lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by
  sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by
  sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by
  sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by
  sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by
  sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by
  sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by
  sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by
  sorry

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := x.root 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_nonneg {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n ≥ 0 := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.root_nonneg {x : Real} (hx : x ≥ 0) {n : ℕ} (hn : n ≥ 1) :
    sSup { y : Real | y ≥ 0 ∧ y ^ n ≤ x } ≥ 0 := by
end TaoBench
import Mathlib.Data.Real.Basic
import Mathlib.Topology.Algebra.Order

/-!
Self-contained slice for Chapter 5.6 target theorem Real.root_of_one.
We avoid importing any Analysis.Section_* modules.
-/

namespace Chapter5

namespace Real

/-- Definition 5.6.2.  We permit ``junk values'' when `x` is negative or `n` vanishes. -/
noncomputable abbrev root (x : Real) (n : ℕ) : Real := sSup { y : Real | y ≥ 0 ∧ y ^ n ≤ x }

end Real

end Chapter5

-- Provide a root-level alias so that dot-notation `(1:Real).root k` resolves.
noncomputable abbrev Real.root (x : Real) (n : ℕ) : Real := Chapter5.Real.root x n

namespace Chapter5

/-- Lemma 5.6.6 (e) / Exercise 5.6.1 -/
theorem Real.root_of_one {k : ℕ} (hk : k ≥ 1) : (1 : Real).root k = 1 := by
end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench
theorem Real.root_of_one {k : ℕ} (hk : k ≥ 1) :
    sSup { y : ℝ | 0 ≤ y ∧ y ^ k ≤ (1 : ℝ) } = 1 := by
end TaoBench
import Mathlib.Data.Real.Basic

namespace Real

noncomputable abbrev root (x : Real) (n : ℕ) : Real := sSup { y : Real | y ≥ 0 ∧ y ^ n ≤ x }

end Real

namespace Chapter5

/-- Lemma 5.6.6 (c) / Exercise 5.6.1 -/
theorem Real.root_pos {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n > 0 ↔ x > 0 := by
end Chapter5
import Mathlib
import Analysis.Section_5_5
import Analysis.Tags.Section_5_5

namespace Chapter5

-- <CONTEXT>
lemma Real.pow_zero (x: Real) : x ^ 0 = 1 := by
  sorry

lemma Real.pow_succ (x: Real) (n:ℕ) : x ^ (n+1) = (x ^ n) * x := by
  sorry

lemma Real.pow_of_coe (q: ℚ) (n:ℕ) : (q:Real) ^ n = (q ^ n:ℚ) := by
  sorry

theorem Real.pow_add (x:Real) (m n:ℕ) : x^n * x^m = x^(n+m) := by
  sorry

theorem Real.pow_mul (x:Real) (m n:ℕ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_pow (x y:Real) (n:ℕ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.pow_eq_zero (x:Real) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by
  sorry

theorem Real.pow_nonneg {x:Real} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by
  sorry

theorem Real.pow_pos {x:Real} (n:ℕ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.pow_ge_pow (x y:Real) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by
  sorry

theorem Real.pow_gt_pow (x y:Real) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by
  sorry

theorem Real.pow_abs (x:Real) (n:ℕ) : |x|^n = |x^n| := by
  sorry

lemma Real.pow_eq_pow (x: Real) (n:ℕ): x ^ (n:ℤ) = x ^ n := by
  sorry

@[simp]
lemma Real.zpow_zero (x: Real) : x ^ (0:ℤ) = 1 := by
  sorry

lemma Real.zpow_neg {x:Real} (n:ℕ) : x^(-n:ℤ) = 1 / (x^n) := by
  sorry

theorem Real.zpow_add (x:Real) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by
  sorry

theorem Real.zpow_mul (x:Real) (n m:ℤ) : (x^n)^m = x^(n*m) := by
  sorry

theorem Real.mul_zpow (x y:Real) (n:ℤ) : (x*y)^n = x^n * y^n := by
  sorry

theorem Real.zpow_pos {x:Real} (n:ℤ) (hx: x > 0) : x^n > 0 := by
  sorry

theorem Real.zpow_ge_zpow {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by
  sorry

theorem Real.zpow_ge_zpow_ofneg {x y:Real} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n < 0) : x^n ≤ y^n := by
  sorry

theorem Real.zpow_inj {x y:Real} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

theorem Real.zpow_abs (x:Real) (n:ℤ) : |x|^n = |x^n| := by
  sorry

noncomputable abbrev _root_.Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

noncomputable abbrev Real.sqrt (x:Real) := _root_.Real.root x 2

theorem Real.rootset_nonempty {x:Real} (hx: x ≥ 0) (n:ℕ) (hn: n ≥ 1) : { y:Real | y ≥ 0 ∧ y^n ≤ x }.Nonempty := by
  sorry

theorem Real.rootset_bddAbove {x:Real} (n:ℕ) (hn: n ≥ 1) : BddAbove { y:Real | y ≥ 0 ∧ y^n ≤ x } := by
  sorry
-- </CONTEXT>
-- <TARGET_THEOREM>
theorem Real.root_pos {x:Real} (hx: x ≥ 0) {n:ℕ} (hn: n ≥ 1) : x.root n > 0 ↔ x > 0 := by
-- </TARGET_THEOREM>

end Chapter5
import Mathlib
import Aesop
set_option maxHeartbeats 0
open BigOperators Real Nat Topology Rat Set Filter

namespace TaoBench

namespace Chapter5

/-- Lemma 5.6.6 (c) / Exercise 5.6.1 -/
theorem Real.root_pos {x : ℝ} (hx : x ≥ 0) {n : ℕ} (hn : n ≥ 1) :
    sSup { y : ℝ | y ≥ 0 ∧ y ^ n ≤ x } > 0 ↔ x > 0 := by
end Chapter5

end TaoBench
import Mathlib.Data.Real.Basic

noncomputable abbrev Real.root (x:Real) (n:ℕ) : Real := sSup { y:Real | y ≥ 0 ∧ y^n ≤ x }

namespace Chapter5

/-- Lemma 5.6.6 (g) / Exercise 5.6.1 -/
theorem Real.root_root {x:Real} (hx: x ≥ 0) {n m:ℕ} (hn: n ≥ 1) (hm: m ≥ 1): (x.root n).root m = x.root (n*m) := by
end Chapter5