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fix: disambiguate Chapter 5.4/5.6 Verso exercise labels #616

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teorth merged 4 commits into from
Aug 2, 2026
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fix: disambiguate Chapter 5.4/5.6 Verso exercise labels #616

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64c47bd
fix: disambiguate Verso labels for Exercises 5.4.6–5.4.8
Chessing234 Aug 2, 2026
96cc917
fix: label Exercise 5.4.9 max identities as parts (a)–(f)
Chessing234 Aug 2, 2026
ef2a70f
fix: disambiguate Verso labels for Exercise 5.6.5 rational powers
Chessing234 Aug 2, 2026
4735ee8
docs: tie negative-z follow-up note to Exercise 5.4.9 (f)
Chessing234 Aug 2, 2026
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38 changes: 19 additions & 19 deletions Analysis/Section_5_4.lean
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Expand Up @@ -329,67 +329,67 @@ theorem Real.floor_exist (x:Real) : ∃! n:ℤ, (n:Real) ≤ x ∧ x < (n:Real)+
/-- Exercise 5.4.4 -/
theorem Real.exist_inv_nat_le {x:Real} (hx: x.IsPos) : ∃ N:ℤ, N>0 ∧ (N:Real)⁻¹ < x := by sorry

/-- Exercise 5.4.6 -/
/-- Exercise 5.4.6 (a) -/
theorem Real.dist_lt_iff (ε x y:Real) : |x-y| < ε ↔ y-ε < x ∧ x < y+ε := by sorry

/-- Exercise 5.4.6 -/
/-- Exercise 5.4.6 (b) -/
theorem Real.dist_le_iff (ε x y:Real) : |x-y| ≤ ε ↔ y-ε ≤ x ∧ x ≤ y+ε := by sorry

/-- Exercise 5.4.7 -/
/-- Exercise 5.4.7 (a) -/
theorem Real.le_add_eps_iff (x y:Real) : (∀ ε > 0, x ≤ y+ε) ↔ x ≤ y := by sorry

/-- Exercise 5.4.7 -/
/-- Exercise 5.4.7 (b) -/
theorem Real.dist_le_eps_iff (x y:Real) : (∀ ε > 0, |x-y| ≤ ε) ↔ x = y := by sorry

/-- Exercise 5.4.8 -/
/-- Exercise 5.4.8 (a) -/
theorem Real.LIM_of_le {x:Real} {a:ℕ → ℚ} (hcauchy: (a:Sequence).IsCauchy) (h: ∀ n, a n ≤ x) :
LIM a ≤ x := by sorry

/-- Exercise 5.4.8 -/
/-- Exercise 5.4.8 (b) -/
theorem Real.LIM_of_ge {x:Real} {a:ℕ → ℚ} (hcauchy: (a:Sequence).IsCauchy) (h: ∀ n, a n ≥ x) :
LIM a ≥ x := by sorry

theorem Real.max_eq (x y:Real) : max x y = if x ≥ y then x else y := max_def' x y

theorem Real.min_eq (x y:Real) : min x y = if x ≤ y then x else y := rfl

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (a) -/
theorem Real.neg_max (x y:Real) : max x y = - min (-x) (-y) := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (b) -/
theorem Real.neg_min (x y:Real) : min x y = - max (-x) (-y) := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (c) -/
theorem Real.max_comm (x y:Real) : max x y = max y x := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (d) -/
theorem Real.max_self (x:Real) : max x x = x := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (e) -/
theorem Real.max_add (x y z:Real) : max (x + z) (y + z) = max x y + z := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (f) -/
theorem Real.max_mul (x y :Real) {z:Real} (hz: z.IsPos) : max (x * z) (y * z) = max x y * z := by
sorry
/- Additional exercise: What happens if z is negative? -/
/- Additional exercise (after 5.4.9 (f)): What happens if z is negative? -/

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (g) -/
theorem Real.min_comm (x y:Real) : min x y = min y x := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (h) -/
theorem Real.min_self (x:Real) : min x x = x := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (i) -/
theorem Real.min_add (x y z:Real) : min (x + z) (y + z) = min x y + z := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (j) -/
theorem Real.min_mul (x y :Real) {z:Real} (hz: z.IsPos) : min (x * z) (y * z) = min x y * z := by
sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (k) -/
theorem Real.inv_max {x y :Real} (hx:x.IsPos) (hy:y.IsPos) : (max x y)⁻¹ = min x⁻¹ y⁻¹ := by sorry

/-- Exercise 5.4.9 -/
/-- Exercise 5.4.9 (l) -/
theorem Real.inv_min {x y :Real} (hx:x.IsPos) (hy:y.IsPos) : (min x y)⁻¹ = max x⁻¹ y⁻¹ := by sorry

/-- Not from textbook: the rationals map as an ordered ring homomorphism into the reals. -/
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4 changes: 2 additions & 2 deletions Analysis/Section_5_6.lean
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Expand Up @@ -248,12 +248,12 @@ theorem Real.ratPow_mul {x y:Real} (hx: x > 0) (hy: y > 0) (q:ℚ) : (x*y)^q = x
/-- Exercise 5.6.3 -/
theorem Real.pow_even (x:Real) {n:ℕ} (hn: Even n) : x^n ≥ 0 := by sorry

/-- Exercise 5.6.5 -/
/-- Exercise 5.6.5 (a) -/
theorem Real.max_ratPow {x y:Real} (hx: x > 0) (hy: y > 0) {q:ℚ} (hq: q > 0) :
max (x^q) (y^q) = (max x y)^q := by
sorry

/-- Exercise 5.6.5 -/
/-- Exercise 5.6.5 (b) -/
theorem Real.min_ratPow {x y:Real} (hx: x > 0) (hy: y > 0) {q:ℚ} (hq: q > 0) :
min (x^q) (y^q) = (min x y)^q := by
sorry
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