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

theorem Real.root_one {x:Real} (hx: x > 0): x.root 1 = x := by
  sorry

theorem Real.pow_cancel {y z:Real} (hy: y > 0) (hz: z > 0) {n:ℕ} (hn: n ≥ 1)
  (h: y^n = z^n) : y = z := by
  sorry

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

theorem Rat.eq_quot (q:ℚ) : ∃ a:ℤ, ∃ b:ℕ, b > 0 ∧ q = a / b := by
  sorry

theorem Real.pow_root_eq_pow_root {a a':ℤ} {b b':ℕ} (hb: b > 0) (hb' : b' > 0)
  (hq : (a/b:ℚ) = (a'/b':ℚ)) {x:Real} (hx: x > 0) :
    (x.root b')^(a') = (x.root b)^(a) := by
  sorry

theorem Real.ratPow_def {x:Real} (hx: x > 0) (a:ℤ) {b:ℕ} (hb: b > 0) : x^(a/b:ℚ) = (x.root b)^a := by
  sorry

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

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

end Chapter5