← Session 17 — Lean tier (i): elementary proofs (after crash)
thinking (3008 chars)
Two mismatches in one: (1) function `fun t => Real.log (t+2))⁻¹` vs `(Real.log (t+2))⁻¹` — the .inv produces `fun t => (f t)⁻¹` where the ⁻¹ is OUTSIDE the lambda body (f t)⁻¹ — parsing: `Real.log (t + 2))⁻¹` — hmm actually it shows `HasDerivAt (fun t => Real.log (t + 2))⁻¹` — the ⁻¹ applies to the FUNCTION?? No — pretty-print ambiguity: `.inv` gives `fun x => (f x)⁻¹` which prints as `(fun t => Real.log (t+2))⁻¹`?? That would be function-inverse. Actually HasDerivAt.inv's conclusion: `HasDerivAt (fun x => (f x)⁻¹) (-f' / f x ^ 2) x` — prints as fun t => (Real.log (t+2))⁻¹ — should be the same as mine... The print shows `HasDerivAt (fun t => Real.log (t + 2))⁻¹` — with the ) before ⁻¹ AFTER the lambda?? ambiguous printing. And (2) derivative: `-(T + 2)⁻¹ / L1 ^ 2` vs `-((T + 2)⁻¹ / L1 ^ 2)` — neg_div vs neg-of-div — defeq, not syntactic.
Fix both with a convert-free massage: take h := hin.inv ... then `convert h using 2` — goals: function eq (rfl? `fun t => (log(t+2))⁻¹` vs h's — might be syntactically equal and convert using 2 handles) and derivative eq: -(T+2)⁻¹/L1² = -((T+2)⁻¹/L1²) by ring. But actually simpler: state hu with EXACTLY h's type: derivative `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2` (parse: (-(T+2)⁻¹)/(L1²)) — wait that's what the error says h HAS: `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2`. So set hu's derivative to `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2` — hmm earlier when I wrote exactly that, the mismatch was the neg placement the other way... Earlier error (line 531 previous round): h had `-((T+2)⁻¹ / L1^2)` expected `-(T+2)⁻¹/L1^2`. NOW h has `-(T+2)⁻¹ / L1^2` expected `-((T+2)⁻¹/L1^2)`?! The difference: previously I used `rwa [e] at h` where e rewrote h's derivative... no wait — previously h came from `hin.inv` directly and printed as `-((T + 2)⁻¹ / Real.log (T + 2) ^ 2)`. Now `hin.inv (ne_of_gt hL1)` prints as `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2`. Same expression, different print?? Odd — maybe Lean normalized differently... whatever: fix by making hu's statement copy EXACTLY what .inv produces. The .inv lemma statement: `HasDerivAt.inv (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun x => (f x)⁻¹) (-f' / f x ^ 2) x`. So derivative is `-f' / f x ^ 2` = `-(T+2)⁻¹ / (Real.log (T+2))^2` — write hu's type as:
(-(T + 2)⁻¹ / Real.log (T + 2) ^ 2)
and function as (fun t => (Real.log (t + 2))⁻¹) — but the function mismatch persists: `fun t => (Real.log (t + 2))⁻¹` vs `fun x => (f x)⁻¹` where f := fun t => Real.log (t+2) — after beta: (Real.log (t+2))⁻¹ ✓ should be syntactically equal post-elaboration... The error ALSO flagged function mismatch? It flagged the whole type as one mismatch — maybe only the derivative part mismatched. Let me write hu exactly as .inv's output shape and let exact work:
have hu : HasDerivAt (fun t => (Real.log (t + 2))⁻¹)
(-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) T := hin.inv (ne_of_gt hL1)
If the function-part is fine, this closes. Then e's LHS updates to the new form: `(-(T + 2)⁻¹ / L1^2) * (T + 2)⁻¹ + ...`.
The `.inv` output shape is `-f' / f^2` — matching it exactly.
tool: Edit ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
"old_string": " have hu : HasDerivAt (fun t => (Real.log (t + 2))\u207b\u00b9)\n (-((T + 2)\u207b\u00b9 / Real.log (T + 2) ^ 2)) T := by\n have hin : HasDerivAt (fun t => Real.log (t + 2)) (T + 2)\u207b\u00b9 T := by\n have h := (Real.hasDerivAt_log (show T + 2 \u2260 0 by linarith)).comp T\n ((hasDerivAt_id' T).add_const 2)\n rwa [mul_one] at h\n exact hin.inv (ne_of_gt hL1)\n have hv : HasDerivAt (fun t => (t + 2)\u207b\u00b9) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) \u2260 0)\n have e : (-(1 : \u211d)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have h := hu.mul hv\n have hT2' : (T + 2) \u2260 0 := ne_of_gt hT2\n have hL1' : Real.log (T + 2) \u2260 0 := ne_of_gt hL1\n have e : (-((T + 2)\u207b\u00b9 / Real.log (T + 2) ^ 2)) * (T + 2)\u207b\u00b9 +\n (Real.log (T + 2))\u207b\u00b9 * (-1 / (T + 2) ^ 2) =\n -(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 := by\n field_simp [hT2', hL1']\n ring\n rwa [e] at h",
"new_string": " have hu : HasDerivAt (fun t => (Real.log (t + 2))\u207b\u00b9)\n (-(T + 2)\u207b\u00b9 / Real.log (T + 2) ^ 2) T := by\n have hin : HasDerivAt (fun t => Real.log (t + 2)) (T + 2)\u207b\u00b9 T := by\n have h := (Real.hasDerivAt_log (show T + 2 \u2260 0 by linarith)).comp T\n ((hasDerivAt_id' T).add_const 2)\n rwa [mul_one] at h\n exact hin.inv (ne_of_gt hL1)\n have hv : HasDerivAt (fun t => (t + 2)\u207b\u00b9) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) \u2260 0)\n have e : (-(1 : \u211d)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have h := hu.mul hv\n have hT2' : (T + 2) \u2260 0 := ne_of_gt hT2\n have hL1' : Real.log (T + 2) \u2260 0 := ne_of_gt hL1\n have e : (-(T + 2)\u207b\u00b9 / Real.log (T + 2) ^ 2) * (T + 2)\u207b\u00b9 +\n (Real.log (T + 2))\u207b\u00b9 * (-1 / (T + 2) ^ 2) =\n -(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 := by\n field_simp [hT2', hL1']\n ring\n rwa [e] at h"
}
result 2026-08-20 05:20:17Z → 2026-08-28 22:13:18Z
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white-space: nowrap; }\n.sheet thead th {\n position: sticky; top: 0; z-index: 2;\n background: var(--bg-panel); color: var(--accent);\n border-bottom: 2px solid var(--border); text-align: left;\n}\n.sheet td { border-bottom: 1px solid #1c2431; }\n.sheet td.num { text-align: right; }\n.sheet .rn {\n position: sticky; left: 0; z-index: 1;\n background: var(--bg-panel); color: var(--muted);\n text-align: right; border-right: 1px solid var(--border);\n user-select: none;\n}\n.sheet thead .rn { z-index: 3; }\n.sheet tbody tr:hover td { background: #1d2635; }\n.sheet tbody tr:hover .rn { background: #232f44; }\n.json-box {\n background: #10151f; border: 1px solid var(--border); border-radius: 6px;\n padding: 10px 14px; max-height: 480px; overflow: auto;\n font-family: var(--mono); font-size: 12.5px; white-space: pre-wrap;\n word-break: break-word;\n}\ndetails summary { cursor: pointer; color: var(--accent); }\n.code-box { white-space: pre; word-break: normal; }\ndetails.perim { margin: 8px 0; }\ndetails.perim > summary { line-height: 1.5; }\n.fb-head { display: flex; gap: 10px; align-items: center; flex-wrap: wrap; margin-bottom: 4px; }\n.fb-notes { color: var(--muted); font-size: 13.5px; }\n\n.messages { margin: 10px 0; }\n.msg { padding: 10px 14px; border-radius: 8px; margin: 6px 0; }\n.msg.success { background: #14301f; border: 1px solid var(--ok); }\n.msg.error { background: #331818; border: 1px solid var(--fail); }\n\n.log-row:hover { background: #1d2635; }\n.freshness { font-size: 12.5px; }\n\n/* ---------- conversations ---------- */\n.msgcard {\n background: var(--bg-card); border: 1px solid var(--border);\n border-radius: 10px; padding: 12px 18px; margin: 12px 0;\n}\n.msg-user { border-left: 4px solid var(--accent2); }\n.msg-assistant { border-left: 4px solid var(--accent); }\n.msg-head { display: flex; gap: 12px; align-items: center; flex-wrap: wrap;\n margin-bottom: 8px; font-size: 13px; }\n.msg-id { font-family: var(--mono); color: var(--muted); }\n.role-badge { font-size: 11.5px; 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[{"type":"thinking","thinking":"Two mismatches in one: (1) function `fun t => Real.log (t+2))⁻¹` vs `(Real.log (t+2))⁻¹` — the .inv produces `fun t => (f t)⁻¹` where the ⁻¹ is OUTSIDE the lambda body (f t)⁻¹ — parsing: `Real.log (t + 2))⁻¹` — hmm actually it shows `HasDerivAt (fun t => Real.log (t + 2))⁻¹` — the ⁻¹ applies to the FUNCTION?? No — pretty-print ambiguity: `.inv` gives `fun x => (f x)⁻¹` which prints as `(fun t => Real.log (t+2))⁻¹`?? That would be function-inverse. Actually HasDerivAt.inv's conclusion: `HasDerivAt (fun x => (f x)⁻¹) (-f' / f x ^ 2) x` — prints as fun t => (Real.log (t+2))⁻¹ — should be the same as mine... The print shows `HasDerivAt (fun t => Real.log (t + 2))⁻¹` — with the ) before ⁻¹ AFTER the lambda?? ambiguous printing. And (2) derivative: `-(T + 2)⁻¹ / L1 ^ 2` vs `-((T + 2)⁻¹ / L1 ^ 2)` — neg_div vs neg-of-div — defeq, not syntactic.\n\nFix both with a convert-free massage: take h := hin.inv ... then `convert h using 2` — goals: function eq (rfl? `fun t => (log(t+2))⁻¹` vs h's — might be syntactically equal and convert using 2 handles) and derivative eq: -(T+2)⁻¹/L1² = -((T+2)⁻¹/L1²) by ring. But actually simpler: state hu with EXACTLY h's type: derivative `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2` (parse: (-(T+2)⁻¹)/(L1²)) — wait that's what the error says h HAS: `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2`. So set hu's derivative to `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2` — hmm earlier when I wrote exactly that, the mismatch was the neg placement the other way... Earlier error (line 531 previous round): h had `-((T+2)⁻¹ / L1^2)` expected `-(T+2)⁻¹/L1^2`. NOW h has `-(T+2)⁻¹ / L1^2` expected `-((T+2)⁻¹/L1^2)`?! The difference: previously I used `rwa [e] at h` where e rewrote h's derivative... no wait — previously h came from `hin.inv` directly and printed as `-((T + 2)⁻¹ / Real.log (T + 2) ^ 2)`. Now `hin.inv (ne_of_gt hL1)` prints as `-(T + 2)⁻¹ / Real.log (T + 2) ^ 2`. Same expression, different print?? Odd — maybe Lean normalized differently... whatever: fix by making hu's statement copy EXACTLY what .inv produces. The .inv lemma statement: `HasDerivAt.inv (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun x => (f x)⁻¹) (-f' / f x ^ 2) x`. So derivative is `-f' / f x ^ 2` = `-(T+2)⁻¹ / (Real.log (T+2))^2` — write hu's type as:\n (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2)\nand function as (fun t => (Real.log (t + 2))⁻¹) — but the function mismatch persists: `fun t => (Real.log (t + 2))⁻¹` vs `fun x => (f x)⁻¹` where f := fun t => Real.log (t+2) — after beta: (Real.log (t+2))⁻¹ ✓ should be syntactically equal post-elaboration... The error ALSO flagged function mismatch? It flagged the whole type as one mismatch — maybe only the derivative part mismatched. Let me write hu exactly as .inv's output shape and let exact work:\n\n have hu : HasDerivAt (fun t => (Real.log (t + 2))⁻¹)\n (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) T := hin.inv (ne_of_gt hL1)\n\nIf the function-part is fine, this closes. Then e's LHS updates to the new form: `(-(T + 2)⁻¹ / L1^2) * (T + 2)⁻¹ + ...`.","signature":""},{"type":"text","text":"The `.inv` output shape is `-f' / f^2` — matching it exactly."},{"type":"tool_use","id":"Edit_121","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":" have hu : HasDerivAt (fun t => (Real.log (t + 2))⁻¹)\n (-((T + 2)⁻¹ / Real.log (T + 2) ^ 2)) T := by\n have hin : HasDerivAt (fun t => Real.log (t + 2)) (T + 2)⁻¹ T := by\n have h := (Real.hasDerivAt_log (show T + 2 ≠ 0 by linarith)).comp T\n ((hasDerivAt_id' T).add_const 2)\n rwa [mul_one] at h\n exact hin.inv (ne_of_gt hL1)\n have hv : HasDerivAt (fun t => (t + 2)⁻¹) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) ≠ 0)\n have e : (-(1 : ℝ)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have h := hu.mul hv\n have hT2' : (T + 2) ≠ 0 := ne_of_gt hT2\n have hL1' : Real.log (T + 2) ≠ 0 := ne_of_gt hL1\n have e : (-((T + 2)⁻¹ / Real.log (T + 2) ^ 2)) * (T + 2)⁻¹ +\n (Real.log (T + 2))⁻¹ * (-1 / (T + 2) ^ 2) =\n -(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 := by\n field_simp [hT2', hL1']\n ring\n rwa [e] at h","new_string":" have hu : HasDerivAt (fun t => (Real.log (t + 2))⁻¹)\n (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) T := by\n have hin : HasDerivAt (fun t => Real.log (t + 2)) (T + 2)⁻¹ T := by\n have h := (Real.hasDerivAt_log (show T + 2 ≠ 0 by linarith)).comp T\n ((hasDerivAt_id' T).add_const 2)\n rwa [mul_one] at h\n exact hin.inv (ne_of_gt hL1)\n have hv : HasDerivAt (fun t => (t + 2)⁻¹) (-1 / (T + 2) ^ 2) T := by\n have h := ((hasDerivAt_id' T).add_const 2).inv (by linarith : (T + 2) ≠ 0)\n have e : (-(1 : ℝ)) / (T + 2) ^ 2 = -1 / (T + 2) ^ 2 := by ring\n rwa [e] at h\n have h := hu.mul hv\n have hT2' : (T + 2) ≠ 0 := ne_of_gt hT2\n have hL1' : Real.log (T + 2) ≠ 0 := ne_of_gt hL1\n have e : (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) * (T + 2)⁻¹ +\n (Real.log (T + 2))⁻¹ * (-1 / (T + 2) ^ 2) =\n -(Real.log (T + 2) + 1) / ((T + 2) * Real.log (T + 2)) ^ 2 := by\n field_simp [hT2', hL1']\n ring\n rwa [e] at h"},"extra_content":null}]