Section_4_3.lean

import Mathlib.Tactic

/-!
# Analysis I, Section 4.3: Absolute value and exponentiation

I have attempted to make the translation as faithful a paraphrasing as possible of the original
text. When there is a choice between a more idiomatic Lean solution and a more faithful
translation, I have generally chosen the latter.  In particular, there will be places where the
Lean code could be "golfed" to be more elegant and idiomatic, but I have consciously avoided
doing so.

Main constructions and results of this section:

- Basic properties of absolute value and exponentiation on the rational numbers (here we use the
  Mathlib rational numbers {lean}`ℚ` rather than the Section 4.2 rational numbers).

Note: to avoid notational conflict, we are using the standard Mathlib definitions of absolute
value and exponentiation.  As such, it is possible to solve several of the exercises here rather
easily using the Mathlib API for these operations.  However, the spirit of the exercises is to
solve these instead using the API provided in this section, as well as more basic Mathlib API for
the rational numbers that does not reference either absolute value or exponentiation.

## Tips from past users

Users of the companion who have completed the exercises in this section are welcome to send their tips for future users in this section as PRs.

- (Add tip here)

-/


/--
  This definition needs to be made outside of the Section 4.3 namespace for technical reasons.
-/
def Rat.Close (ε : ℚ) (x y:ℚ) := |x-y| ≤ ε


namespace Section_4_3

/-- Definition 4.3.1 (Absolute value) -/
abbrev abs (x:ℚ) : ℚ := if x > 0 then x else (if x < 0 then -x else 0)

theorem abs_of_pos {x: ℚ} (hx: 0 < x) : abs x = x := by grind

/-- Definition 4.3.1 (Absolute value) -/
theorem abs_of_neg {x: ℚ} (hx: x < 0) : abs x = -x := by grind

/-- Definition 4.3.1 (Absolute value) -/
theorem abs_of_zero : abs 0 = 0 := rfl

/--
  (Not from textbook) This definition of absolute value agrees with the Mathlib one.
  Henceforth we use the Mathlib absolute value.
-/
theorem abs_eq_abs (x: ℚ) : abs x = |x| := by
  sorry

abbrev dist (x y : ℚ) := |x - y|

/--
  Definition 4.2 (Distance).
  We avoid the Mathlib notion of distance here because it is real-valued.
-/
theorem dist_eq (x y: ℚ) : dist x y = |x-y| := rfl

/-- Proposition 4.3.3(a) / Exercise 4.3.1 -/
theorem abs_nonneg (x: ℚ) : |x| ≥ 0 := by sorry

/-- Proposition 4.3.3(a) / Exercise 4.3.1 -/
theorem abs_eq_zero_iff (x: ℚ) : |x| = 0 ↔ x = 0 := by sorry

/-- Proposition 4.3.3(b) / Exercise 4.3.1 -/
theorem abs_add (x y:ℚ) : |x + y| ≤ |x| + |y| := by sorry

/-- Proposition 4.3.3(c) / Exercise 4.3.1 -/
theorem abs_le_iff (x y:ℚ) : -y ≤ x ∧ x ≤ y ↔ |x| ≤ y := by sorry

/-- Proposition 4.3.3(c) / Exercise 4.3.1 -/
theorem le_abs (x:ℚ) : -|x| ≤ x ∧ x ≤ |x| := by sorry

/-- Proposition 4.3.3(d) / Exercise 4.3.1 -/
theorem abs_mul (x y:ℚ) : |x * y| = |x| * |y| := by sorry

/-- Proposition 4.3.3(d) / Exercise 4.3.1 -/
theorem abs_neg (x:ℚ) : |-x| = |x| := by sorry

/-- Proposition 4.3.3(e) / Exercise 4.3.1 -/
theorem dist_nonneg (x y:ℚ) : dist x y ≥ 0 := by sorry

/-- Proposition 4.3.3(e) / Exercise 4.3.1 -/
theorem dist_eq_zero_iff (x y:ℚ) : dist x y = 0 ↔ x = y := by
  sorry

/-- Proposition 4.3.3(f) / Exercise 4.3.1 -/
theorem dist_symm (x y:ℚ) : dist x y = dist y x := by sorry

/-- Proposition 4.3.3(f) / Exercise 4.3.1 -/
theorem dist_le (x y z:ℚ) : dist x z ≤ dist x y + dist y z := by sorry

/--
  Definition 4.3.4 (eps-closeness).  In the text the notion is undefined for ε zero or negative,
  but it is more convenient in Lean to assign a "junk" definition in this case.  But this also
  allows some relaxations of hypotheses in the lemmas that follow.
-/
theorem close_iff (ε x y:ℚ): ε.Close x y ↔ |x - y| ≤ ε := by rfl

/-- Examples 4.3.6 -/
example : (0.1:ℚ).Close (0.99:ℚ) (1.01:ℚ) := by sorry

/-- Examples 4.3.6 -/
example : ¬ (0.01:ℚ).Close (0.99:ℚ) (1.01:ℚ) := by sorry

/-- Examples 4.3.6 -/
example (ε : ℚ) (hε : ε > 0) : ε.Close 2 2 := by sorry

theorem close_refl (x:ℚ) : (0:ℚ).Close x x := by sorry

/-- Proposition 4.3.7(a) / Exercise 4.3.2 -/
theorem eq_if_close (x y:ℚ) : x = y ↔ ∀ ε:ℚ, ε > 0 → ε.Close x y := by sorry

/-- Proposition 4.3.7(b) / Exercise 4.3.2 -/
theorem close_symm (ε x y:ℚ) : ε.Close x y ↔ ε.Close y x := by sorry

/-- Proposition 4.3.7(c) / Exercise 4.3.2 -/
theorem close_trans {ε δ x y z:ℚ} (hxy: ε.Close x y) (hyz: δ.Close y z) :
    (ε + δ).Close x z := by sorry

/-- Proposition 4.3.7(d) / Exercise 4.3.2 -/
theorem add_close {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε + δ).Close (x+z) (y+w) := by sorry

/-- Proposition 4.3.7(d) / Exercise 4.3.2 -/
theorem sub_close {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε + δ).Close (x-z) (y-w) := by sorry

/-- Proposition 4.3.7(e) / Exercise 4.3.2, slightly strengthened -/
theorem close_mono {ε ε' x y:ℚ} (hxy: ε.Close x y) (hε: ε' ≥  ε) :
    ε'.Close x y := by sorry

/-- Proposition 4.3.7(f) / Exercise 4.3.2 -/
theorem close_between {ε x y z w:ℚ} (hxy: ε.Close x y) (hxz: ε.Close x z)
  (hbetween: (y ≤ w ∧ w ≤ z) ∨ (z ≤ w ∧ w ≤ y)) : ε.Close x w := by sorry

/-- Proposition 4.3.7(g) / Exercise 4.3.2 -/
theorem close_mul_right {ε x y z:ℚ} (hxy: ε.Close x y) :
    (ε*|z|).Close (x * z) (y * z) := by sorry

/-- Proposition 4.3.7(h) / Exercise 4.3.2 -/
theorem close_mul_mul {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε*|z|+δ*|x|+ε*δ).Close (x * z) (y * w) := by
  -- The proof is written to follow the structure of the original text, though
  -- non-negativity of ε and δ are implied and don't need to be provided as
  -- explicit hypotheses.
  have hε : ε ≥ 0 := le_trans (abs_nonneg _) hxy
  set a := y-x
  have ha : y = x + a := by grind
  have haε: |a| ≤ ε := by rwa [close_symm, close_iff] at hxy
  set b := w-z
  have hb : w = z + b := by grind
  have hbδ: |b| ≤ δ := by rwa [close_symm, close_iff] at hzw
  have : y*w = x * z + a * z + x * b + a * b := by grind
  rw [close_symm, close_iff]
  calc
    _ = |a * z + b * x + a * b| := by grind
    _ ≤ |a * z + b * x| + |a * b| := abs_add _ _
    _ ≤ |a * z| + |b * x| + |a * b| := by grind [abs_add]
    _ = |a| * |z| + |b| * |x| + |a| * |b| := by grind [abs_mul]
    _ ≤ _ := by gcongr

/-- This variant of Proposition 4.3.7(h) was not in the textbook, but can be useful
in some later exercises. -/
theorem close_mul_mul' {ε δ x y z w:ℚ} (hxy: ε.Close x y) (hzw: δ.Close z w) :
    (ε*|z|+δ*|y|).Close (x * z) (y * w) := by
    sorry

/-- Definition 4.3.9 (exponentiation).  Here we use the Mathlib definition.-/
lemma pow_zero (x:ℚ) : x^0 = 1 := _root_.pow_zero x

example : (0:ℚ)^0 = 1 := pow_zero 0

/-- Definition 4.3.9 (exponentiation).  Here we use the Mathlib definition.-/
lemma pow_succ (x:ℚ) (n:ℕ) : x^(n+1) = x^n * x := _root_.pow_succ x n

/-- Proposition 4.3.10(a) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_add (x:ℚ) (m n:ℕ) : x^n * x^m = x^(n+m) := by sorry

/-- Proposition 4.3.10(a) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_mul (x:ℚ) (m n:ℕ) : (x^n)^m = x^(n*m) := by sorry

/-- Proposition 4.3.10(a) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem mul_pow (x y:ℚ) (n:ℕ) : (x*y)^n = x^n * y^n := by sorry

/-- Proposition 4.3.10(b) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_eq_zero (x:ℚ) (n:ℕ) (hn : 0 < n) : x^n = 0 ↔ x = 0 := by sorry

/-- Proposition 4.3.10(c) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_nonneg {x:ℚ} (n:ℕ) (hx: x ≥ 0) : x^n ≥ 0 := by sorry

/-- Proposition 4.3.10(c) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_pos {x:ℚ} (n:ℕ) (hx: x > 0) : x^n > 0 := by sorry

/-- Proposition 4.3.10(c) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_ge_pow (x y:ℚ) (n:ℕ) (hxy: x ≥ y) (hy: y ≥ 0) : x^n ≥ y^n := by sorry

/-- Proposition 4.3.10(c) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_gt_pow (x y:ℚ) (n:ℕ) (hxy: x > y) (hy: y ≥ 0) (hn: n > 0) : x^n > y^n := by sorry

/-- Proposition 4.3.10(d) (Properties of exponentiation, I) / Exercise 4.3.3 -/
theorem pow_abs (x:ℚ) (n:ℕ) : |x|^n = |x^n| := by sorry

/--
  Definition 4.3.11 (Exponentiation to a negative number).
  Here we use the Mathlib notion of integer exponentiation
-/
theorem zpow_neg (x:ℚ) (n:ℕ) : x^(-(n:ℤ)) = 1/(x^n) := by simp

example (x:ℚ): x^(-3:ℤ) = 1/(x^3) := zpow_neg x 3

example (x:ℚ): x^(-3:ℤ) = 1/(x*x*x) := by convert zpow_neg x 3; ring

theorem pow_eq_zpow (x:ℚ) (n:ℕ): x^(n:ℤ) = x^n := zpow_natCast x n

/-- Proposition 4.3.12(a) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_add (x:ℚ) (n m:ℤ) (hx: x ≠ 0): x^n * x^m = x^(n+m) := by sorry

/-- Proposition 4.3.12(a) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_mul (x:ℚ) (n m:ℤ) : (x^n)^m = x^(n*m) := by sorry

/-- Proposition 4.3.12(a) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem mul_zpow (x y:ℚ) (n:ℤ) : (x*y)^n = x^n * y^n := by sorry

/-- Proposition 4.3.12(b) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_pos {x:ℚ} (n:ℤ) (hx: x > 0) : x^n > 0 := by sorry

/-- Proposition 4.3.12(b) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_ge_zpow {x y:ℚ} {n:ℤ} (hxy: x ≥ y) (hy: y > 0) (hn: n > 0): x^n ≥ y^n := by sorry

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

/-- Proposition 4.3.12(c) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_inj {x y:ℚ} {n:ℤ} (hx: x > 0) (hy : y > 0) (hn: n ≠ 0) (hxy: x^n = y^n) : x = y := by
  sorry

/-- Proposition 4.3.12(d) (Properties of exponentiation, II) / Exercise 4.3.4 -/
theorem zpow_abs (x:ℚ) (n:ℤ) : |x|^n = |x^n| := by sorry

/-- Exercise 4.3.5 -/
theorem two_pow_geq (N:ℕ) : 2^N ≥ N := by sorry