feat(Chapter_02): prove all remaining theorems#111
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pitmonticone wants to merge 2 commits intomo271:mainfrom
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feat(Chapter_02): prove all remaining theorems#111pitmonticone wants to merge 2 commits intomo271:mainfrom
pitmonticone wants to merge 2 commits intomo271:mainfrom
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Co-authored-by: aristotle-harmonic@harmonic.fun. Co-Authored-By: Aristotle <232223898+aristotle-harmonic@users.noreply.github.com>
mo271
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Jan 1, 2026
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thanks, I find these proofs a bit long and suggested shorter ones
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| have h_ln : Real.log n.factorial > 1 + n * Real.log n - n := by | ||
| induction hn | ||
| · have := log_lt_sub_one_of_pos two_pos; norm_num at *; linarith | ||
| · norm_num [factorial_succ] at * | ||
| rw [log_mul (by positivity) (by positivity)] | ||
| have h_log_bound : ∀ m : ℕ, 2 ≤ m → Real.log (m + 1) < Real.log m + 1 / m := by | ||
| intro m hm | ||
| rw [log_lt_iff_lt_exp (by positivity), exp_add, exp_log (by positivity)] | ||
| ring_nf | ||
| nlinarith [add_one_lt_exp (by positivity : (m : ℝ) ⁻¹ ≠ 0), | ||
| (by norm_cast : (2 : ℝ) ≤ m), mul_inv_cancel₀ (by positivity : (m : ℝ) ≠ 0)] | ||
| nlinarith [h_log_bound _ ‹_›, one_div_mul_cancel (by positivity : ((Nat.cast : ℕ → ℝ) ‹_›) ≠ 0)] | ||
| have h_exp : (n.factorial : ℝ) > exp (1 + n * Real.log n - n) := by | ||
| rwa [gt_iff_lt, lt_log_iff_exp_lt (by positivity)] at h_ln | ||
| convert h_exp using 1; norm_num [exp_add, exp_sub, exp_nat_mul, exp_log | ||
| (by positivity : 0 < (n : ℝ)), chapter2.e] | ||
| ring_nf | ||
| norm_num [Real.exp_neg, Real.exp_nat_mul] |
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Suggested change
| have h_ln : Real.log n.factorial > 1 + n * Real.log n - n := by | |
| induction hn | |
| · have := log_lt_sub_one_of_pos two_pos; norm_num at *; linarith | |
| · norm_num [factorial_succ] at * | |
| rw [log_mul (by positivity) (by positivity)] | |
| have h_log_bound : ∀ m : ℕ, 2 ≤ m → Real.log (m + 1) < Real.log m + 1 / m := by | |
| intro m hm | |
| rw [log_lt_iff_lt_exp (by positivity), exp_add, exp_log (by positivity)] | |
| ring_nf | |
| nlinarith [add_one_lt_exp (by positivity : (m : ℝ) ⁻¹ ≠ 0), | |
| (by norm_cast : (2 : ℝ) ≤ m), mul_inv_cancel₀ (by positivity : (m : ℝ) ≠ 0)] | |
| nlinarith [h_log_bound _ ‹_›, one_div_mul_cancel (by positivity : ((Nat.cast : ℕ → ℝ) ‹_›) ≠ 0)] | |
| have h_exp : (n.factorial : ℝ) > exp (1 + n * Real.log n - n) := by | |
| rwa [gt_iff_lt, lt_log_iff_exp_lt (by positivity)] at h_ln | |
| convert h_exp using 1; norm_num [exp_add, exp_sub, exp_nat_mul, exp_log | |
| (by positivity : 0 < (n : ℝ)), chapter2.e] | |
| ring_nf | |
| norm_num [Real.exp_neg, Real.exp_nat_mul] | |
| refine exp_log (by positivity : rexp 1 * (n / rexp 1)^n > 0) ▸ exp_log.comp Nat.cast_pos.2 n.factorial_pos ▸ Nat.le_induction (?_) ? _ _ hn | |
| · norm_num [(by linear_combination log_two_lt_d9: 1 + (2 * (log 2 - 1)) < log 2), log_mul, log_div] | |
| · field_simp [log_mul,mul_left_comm, log_div, Nat.factorial, pow_succ'] | |
| intro n hn h | |
| linear_combination h + ↑n * (by field_simp [←log_div _,((log_le_sub_one_of_pos _)).trans] : log (n + 1)-log n ≤ 1/n) + div_self_le_one (n: ℝ) |
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@mo271 The golfed proof doesn't compile as it is.
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| constructor | ||
| · induction hn <;> norm_num [harmonic] at * | ||
| · norm_num [Finset.sum_range_succ] | ||
| linarith [@Real.log_lt_sub_one_of_pos 2 zero_lt_two (OfNat.one_ne_ofNat 2).symm] | ||
| · rw [Finset.sum_range_succ] | ||
| rename_i k hk ih | ||
| rw [show (k : ℝ) + 1 = k * (1 + (k : ℝ) ⁻¹) by | ||
| nlinarith only [mul_inv_cancel₀ (by positivity : (k : ℝ) ≠ 0)], | ||
| Real.log_mul (by positivity) (by positivity)] | ||
| nlinarith [inv_pos.mpr (by positivity : 0 < (k : ℝ)), | ||
| inv_pos.mpr (by positivity : 0 < (k : ℝ) * (1 + (k : ℝ) ⁻¹)), mul_inv_cancel₀ | ||
| (by positivity : (k : ℝ) ≠ 0), mul_inv_cancel₀ | ||
| (by positivity : (k : ℝ) * (1 + (k : ℝ) ⁻¹) ≠ 0), Real.log_lt_sub_one_of_pos | ||
| (by positivity : 0 < (1 + (k : ℝ) ⁻¹)) | ||
| (by nlinarith [inv_mul_cancel₀ (by positivity : (k : ℝ) ≠ 0)])] | ||
| · have h_harmonic : ∀ n : ℕ, 1 < n → (∑ k ∈ Finset.range n, (1 / (k + 1) : ℝ)) < Real.log n + 1 := by | ||
| intro n hn | ||
| have h_harmonic : ∀ k ∈ Finset.range n, k ≠ 0 → (1 / (k + 1) : ℝ) < Real.log (k + 1) - Real.log k := by | ||
| intro k hk hk'; rw [← Real.log_div (by positivity) (by positivity)]; ring_nf | ||
| rw [mul_inv_cancel₀ (by positivity), inv_eq_one_div, div_lt_iff₀] <;> nlinarith [Real.log_inv (1 + (k : ℝ) ⁻¹), | ||
| Real.log_lt_sub_one_of_pos (inv_pos.mpr (by positivity : 0 < (1 + (k : ℝ) ⁻¹))) | ||
| (by aesop), inv_pos.mpr (by positivity : 0 < (1 + (k : ℝ) ⁻¹)), mul_inv_cancel₀ | ||
| (by positivity : (1 + (k : ℝ) ⁻¹) ≠ 0), inv_pos.mpr (by positivity : 0 < (k : ℝ)), | ||
| mul_inv_cancel₀ (by positivity : (k : ℝ) ≠ 0)] | ||
| have h_sum : ∑ k ∈ Finset.range n, (1 / (k + 1) : ℝ) < 1 + ∑ k ∈ Finset.range (n - 1), (Real.log (k + 2) - Real.log (k + 1)) := by | ||
| rcases n with ⟨⟩ <;> norm_num [add_comm, add_left_comm, Finset.sum_range_succ'] at * | ||
| rw [← Finset.sum_sub_distrib] | ||
| exact Finset.sum_lt_sum_of_nonempty ⟨_, Finset.mem_range.mpr hn⟩ fun x hx ↦ by | ||
| have := h_harmonic (x + 1) (by linarith [Finset.mem_range.mp hx]) | ||
| (by linarith [Finset.mem_range.mp hx]); norm_num [add_assoc] at *; linarith | ||
| have h_telescope : | ||
| ∑ k ∈ Finset.range (n - 1), | ||
| (Real.log (k + 2) - Real.log (k + 1)) = Real.log n - Real.log 1 := by | ||
| exact_mod_cast Eq.trans (Finset.sum_range_sub (fun k ↦ Real.log (k + 1)) _) | ||
| (by cases n <;> norm_num at *) | ||
| norm_num at *; linarith | ||
| simpa [harmonic] using h_harmonic n hn |
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The entier proof could be golfed:
use(2).le_induction ?_ ? _ _ hn, (2).le_induction ?_ ? _ _ hn
· norm_num [harmonic, add_lt_of_lt_sub_right ∘ log_two_lt_d9.trans]
· push_cast [harmonic_succ, one_div]
intro n hn h
apply add_lt_add_right (h.trans_le'.comp (sub_le_iff_le_add').mp (by field_simp [← log_div _, (( Real.log_le_sub_one_of_pos _)).trans]))
· norm_num [harmonic, ←sub_lt_iff_lt_add, log_two_gt_d9.trans']
· intro n hn h
apply lt_of_le_of_lt (by rw [harmonic_succ, Rat.cast_add]) ((add_lt_add_right h _).trans_le.comp (add_right_comm _ _ _).trans_le (add_right_mono _))
exact le_sub_iff_add_le'.1 (by field_simp [le_log_iff_exp_le, (exp_bound_div_one_sub_of_interval' _ _).le.trans, div_lt_self, lt_add_of_pos_left, ←log_div])
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@mo271 The golfed proof doesn't compile as it is.
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In general in the end the proofs should not just work, but follow the informal proofs in the Book... |
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But it is ok to fill them in now, perhaps I should add a TODO to make them more align with the informal arguments... |
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Co-authored-by: aristotle-harmonic@harmonic.fun.