← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (498 chars)
`Real.exp_half (x : ℝ) : Real.exp (x / 2) = √(Real.exp x)` — in the Real namespace presumably (Real.exp_half). So the rewrite: goal `√(Real.exp 5) = Real.exp (5 / 2)`: `rw [← Real.exp_half]` — rewrites exp(x/2) → √(exp x)... the goal's RHS is exp(5/2): ← Real.exp_half rewrites it to √(exp 5) ✓ both sides equal ✓. Or prove the step as `(Real.exp_half 5).symm`. Use:
```
_ = Real.exp (5 / 2) := (Real.exp_half 5).symm
```
Clean. Also fix the sqrt_div pinning. Let me apply both edits.
`Real.exp_half` is the name. Applying both fixes.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " rw [norm_div, hnormA, hnormB, \u2190 Real.sqrt_div hQpos]",
"new_string": " rw [norm_div, hnormA, hnormB,\n \u2190 Real.sqrt_div (by positivity : (0 : \u211d) \u2264 9 + t ^ 2)]"
}
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
"old_string": " calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := by rw [Real.sqrt_exp]",
"new_string": " calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm"
}
raw content_json (database row)
[{"type":"thinking","thinking":"`Real.exp_half (x : ℝ) : Real.exp (x / 2) = √(Real.exp x)` — in the Real namespace presumably (Real.exp_half). So the rewrite: goal `√(Real.exp 5) = Real.exp (5 / 2)`: `rw [← Real.exp_half]` — rewrites exp(x/2) → √(exp x)... the goal's RHS is exp(5/2): ← Real.exp_half rewrites it to √(exp 5) ✓ both sides equal ✓. Or prove the step as `(Real.exp_half 5).symm`. Use:\n```\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm\n```\nClean. Also fix the sqrt_div pinning. Let me apply both edits.","signature":""},{"type":"text","text":"`Real.exp_half` is the name. Applying both fixes."},{"type":"tool_use","id":"Edit_211","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" rw [norm_div, hnormA, hnormB, ← Real.sqrt_div hQpos]","new_string":" rw [norm_div, hnormA, hnormB,\n ← Real.sqrt_div (by positivity : (0 : ℝ) ≤ 9 + t ^ 2)]"},"extra_content":null},{"type":"tool_use","id":"Edit_212","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":" calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n ≤ Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := by rw [Real.sqrt_exp]","new_string":" calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n ≤ Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n _ = Real.exp (5 / 2) := (Real.exp_half 5).symm"},"extra_content":null}]