← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (1904 chars)
No Int.toNat_mono in this Mathlib. For `2 ≤ m.toNat` from `2 ≤ m` (m : ℤ): simplest: `Int.toNat_le_toNat`? no — different typeclass. Use omega: `Int.toNat_of_nonneg`: for 0 ≤ m: m = ↑(m.toNat): so 2 ≤ m = ↑(m.toNat) ⟹ 2 ≤ m.toNat via exact_mod_cast:
```
have hn2 : 2 ≤ m.toNat := by
have h1 : (m : ℤ) = ((m.toNat : ℕ) : ℤ) := (Int.toNat_of_nonneg (by omega)).symm
have h2 : (2 : ℤ) ≤ ((m.toNat : ℕ) : ℤ) := by rwa [← h1]
exact_mod_cast h2
```
Hmm — Int.toNat_of_nonneg (h : 0 ≤ m) : ↑(m.toNat) = m — so h1 : m = ↑(m.toNat) via .symm ✓; then 2 ≤ ↑(m.toNat) via rwa [← h1] at hm2-ish: `have h2 : (2:ℤ) ≤ ((m.toNat:ℕ):ℤ) := by rw [h1] at hm2; exact hm2`? careful: rw [← h1] rewrites m → ↑(m.toNat)... in hm2 : 2 ≤ m: rw [← h1] at hm2: h1 : m = ↑(m.toNat); ← h1 : ↑(m.toNat) = m — rewrites ↑(m.toNat) to m — wrong direction. rw [h1] at hm2 rewrites m → ↑(m.toNat): hm2 : 2 ≤ ↑(m.toNat) ✓. Then exact_mod_cast hm2 : 2 ≤ m.toNat ✓:
```
have hn2 : 2 ≤ m.toNat := by
have h1 : (m : ℤ) = ((m.toNat : ℕ) : ℤ) := (Int.toNat_of_nonneg (by omega)).symm
rw [h1] at hm2
exact_mod_cast hm2
```
Wait — rw [h1] at hm2 rewrites ALL occurrences of m in hm2 — hm2 : 2 ≤ m → 2 ≤ ↑(m.toNat) ✓ (m only there). But careful: h1's LHS m also appears in... only hm2. ✓. And hn2's goal 2 ≤ m.toNat: exact_mod_cast hm2 : hm2 : (2:ℤ) ≤ ↑(m.toNat): → (2:ℕ) ≤ m.toNat via exact_mod_cast ✓.
Similarly hwn's exact_mod_cast direction fix (.symm).
Let me now apply all fixes:
1. Line 44: remove `simp only at h`.
2. hlim0: refine squeeze_zero with named f g.
3. digamma_natCast_eq h4: add the final `rw [harmonic]` for the RHS unfold.
4. summable_digamma_majorant line ~144: ascribe the positivity.
5. norm_digamma_term_le: ascribe mul_pos parts.
6. sum_one_div_le_one_add_log: add Nat.cast_add_one to the rw.
7. hw_norm_ge: linarith finish.
8. hn2: the Int.toNat_of_nonneg route.
9. hwn: .symm.
Applying the batch of fixes.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean",
"old_string": " have h := Finset.sum_range_sub' (fun n : \u2115 => (1 : \u211d) / ((n : \u211d) + N)) M\n simp only at h\n rw [Nat.cast_zero, zero_add] at h",
"new_string": " have h := Finset.sum_range_sub' (fun n : \u2115 => (1 : \u211d) / ((n : \u211d) + N)) M\n rw [Nat.cast_zero, zero_add] at h"
}
raw content_json (database row)
[{"type":"thinking","thinking":"No Int.toNat_mono in this Mathlib. For `2 ≤ m.toNat` from `2 ≤ m` (m : ℤ): simplest: `Int.toNat_le_toNat`? no — different typeclass. Use omega: `Int.toNat_of_nonneg`: for 0 ≤ m: m = ↑(m.toNat): so 2 ≤ m = ↑(m.toNat) ⟹ 2 ≤ m.toNat via exact_mod_cast:\n```\n have hn2 : 2 ≤ m.toNat := by\n have h1 : (m : ℤ) = ((m.toNat : ℕ) : ℤ) := (Int.toNat_of_nonneg (by omega)).symm\n have h2 : (2 : ℤ) ≤ ((m.toNat : ℕ) : ℤ) := by rwa [← h1]\n exact_mod_cast h2\n```\nHmm — Int.toNat_of_nonneg (h : 0 ≤ m) : ↑(m.toNat) = m — so h1 : m = ↑(m.toNat) via .symm ✓; then 2 ≤ ↑(m.toNat) via rwa [← h1] at hm2-ish: `have h2 : (2:ℤ) ≤ ((m.toNat:ℕ):ℤ) := by rw [h1] at hm2; exact hm2`? careful: rw [← h1] rewrites m → ↑(m.toNat)... in hm2 : 2 ≤ m: rw [← h1] at hm2: h1 : m = ↑(m.toNat); ← h1 : ↑(m.toNat) = m — rewrites ↑(m.toNat) to m — wrong direction. rw [h1] at hm2 rewrites m → ↑(m.toNat): hm2 : 2 ≤ ↑(m.toNat) ✓. Then exact_mod_cast hm2 : 2 ≤ m.toNat ✓:\n```\n have hn2 : 2 ≤ m.toNat := by\n have h1 : (m : ℤ) = ((m.toNat : ℕ) : ℤ) := (Int.toNat_of_nonneg (by omega)).symm\n rw [h1] at hm2\n exact_mod_cast hm2\n```\nWait — rw [h1] at hm2 rewrites ALL occurrences of m in hm2 — hm2 : 2 ≤ m → 2 ≤ ↑(m.toNat) ✓ (m only there). But careful: h1's LHS m also appears in... only hm2. ✓. And hn2's goal 2 ≤ m.toNat: exact_mod_cast hm2 : hm2 : (2:ℤ) ≤ ↑(m.toNat): → (2:ℕ) ≤ m.toNat via exact_mod_cast ✓.\n\nSimilarly hwn's exact_mod_cast direction fix (.symm).\n\nLet me now apply all fixes:\n1. Line 44: remove `simp only at h`.\n2. hlim0: refine squeeze_zero with named f g.\n3. digamma_natCast_eq h4: add the final `rw [harmonic]` for the RHS unfold.\n4. summable_digamma_majorant line ~144: ascribe the positivity.\n5. norm_digamma_term_le: ascribe mul_pos parts.\n6. sum_one_div_le_one_add_log: add Nat.cast_add_one to the rw.\n7. hw_norm_ge: linarith finish.\n8. hn2: the Int.toNat_of_nonneg route.\n9. hwn: .symm.","signature":""},{"type":"text","text":"Applying the batch of fixes."},{"type":"tool_use","id":"Edit_283","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean","old_string":" have h := Finset.sum_range_sub' (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + N)) M\n simp only at h\n rw [Nat.cast_zero, zero_add] at h","new_string":" have h := Finset.sum_range_sub' (fun n : ℕ => (1 : ℝ) / ((n : ℝ) + N)) M\n rw [Nat.cast_zero, zero_add] at h"},"extra_content":null}]