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

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_of_one {k : ℕ} (hk : k ≥ 1) : (1 : Real).root k = 1 := by
  sorry
-- </TARGET_THEOREM>

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
end Chapter5