← Session 18 — Lean tier (ii): finite-height proof + certified base points
thinking (1335 chars)
All 8 updated. Now add sInftyLower def + refactor sInfty_lower_bound's statement. The current statement:
```
theorem sInfty_lower_bound (n : ℕ) (hn : 20 ≤ n) :
(1 / 2) * n * Real.log n +
(lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4 ≤
SInfty n := by
```
New: define sInftyLower before the theorem (near SInfty def or right before sInfty_lower_bound). I'll add the def right after lagariasC1 (line ~76), then change the theorem statement to `sInftyLower n ≤ SInfty n` and add a `show` at proof start.
Proof starts: `have hmain : ∀ K : ℕ, (1/2)*(n:ℝ)*Real.log n + ... - 4 ≤ ... → ... ≤ SInfty n` — the hmain hypothesis uses the explicit expression. Then at the end `refine hmain (q-1) ?_` etc. If I change the goal via `show (1/2)*(n:ℝ)*... ≤ SInfty n` — since sInftyLower ↑n is definitionally that expression, `show` works.
Wait — careful with the def: sInftyLower (n : ℝ) : ℝ := (1/2)*n*Real.log n + (lagariasC1 - Real.eulerMascheroniConstant/2)*n - 4. Then sInftyLower (↑n) = (1/2)*↑n*log ↑n + ... The theorem's current expression: `(1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4` with n : ℕ coerced. `show` should unify: sInftyLower ↑n unfolds to (1/2)*↑n*Real.log ↑n + ... — same as goal's coerced expression ✓ (Real.log n is Real.log ↑n).
Add def + edit theorem.
Now add `sInftyLower` and re-express `sInfty_lower_bound` in terms of it.
tool: Edit
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"old_string": "/-- C\u2081 = (\u03b3 \u2212 1 \u2212 log 2\u03c0)/2 (D-037). -/\ndef lagariasC1 : \u211d :=\n (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2",
"new_string": "/-- C\u2081 = (\u03b3 \u2212 1 \u2212 log 2\u03c0)/2 (D-037). -/\ndef lagariasC1 : \u211d :=\n (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2\n\n/-- The explicit archimedean lower bound S(n) = \u00bdn log n + (C\u2081\u2212\u03b3/2)n \u2212 4\n(D-037 (3.1)); named so the assembly can refer to it. -/\ndef sInftyLower (n : \u211d) : \u211d :=\n (1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4"
}
raw content_json (database row)
[{"type":"thinking","thinking":"All 8 updated. Now add sInftyLower def + refactor sInfty_lower_bound's statement. The current statement:\n\n```\ntheorem sInfty_lower_bound (n : ℕ) (hn : 20 ≤ n) :\n (1 / 2) * n * Real.log n +\n (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4 ≤\n SInfty n := by\n```\n\nNew: define sInftyLower before the theorem (near SInfty def or right before sInfty_lower_bound). I'll add the def right after lagariasC1 (line ~76), then change the theorem statement to `sInftyLower n ≤ SInfty n` and add a `show` at proof start.\n\nProof starts: `have hmain : ∀ K : ℕ, (1/2)*(n:ℝ)*Real.log n + ... - 4 ≤ ... → ... ≤ SInfty n` — the hmain hypothesis uses the explicit expression. Then at the end `refine hmain (q-1) ?_` etc. If I change the goal via `show (1/2)*(n:ℝ)*... ≤ SInfty n` — since sInftyLower ↑n is definitionally that expression, `show` works.\n\nWait — careful with the def: sInftyLower (n : ℝ) : ℝ := (1/2)*n*Real.log n + (lagariasC1 - Real.eulerMascheroniConstant/2)*n - 4. Then sInftyLower (↑n) = (1/2)*↑n*log ↑n + ... The theorem's current expression: `(1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4` with n : ℕ coerced. `show` should unify: sInftyLower ↑n unfolds to (1/2)*↑n*Real.log ↑n + ... — same as goal's coerced expression ✓ (Real.log n is Real.log ↑n).\n\nAdd def + edit theorem.","signature":""},{"type":"text","text":"Now add `sInftyLower` and re-express `sInfty_lower_bound` in terms of it."},{"type":"tool_use","id":"Edit_43","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- C₁ = (γ − 1 − log 2π)/2 (D-037). -/\ndef lagariasC1 : ℝ :=\n (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2","new_string":"/-- C₁ = (γ − 1 − log 2π)/2 (D-037). -/\ndef lagariasC1 : ℝ :=\n (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2\n\n/-- The explicit archimedean lower bound S(n) = ½n log n + (C₁−γ/2)n − 4\n(D-037 (3.1)); named so the assembly can refer to it. -/\ndef sInftyLower (n : ℝ) : ℝ :=\n (1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4"},"extra_content":null}]