[{"type":"thinking","thinking":"`AntitoneOn.summable_of_integrableOn_Ioi {N : ℕ} (anti : AntitoneOn f (Ici (N : ℝ))) (integrable : IntegrableOn f (Ioi (N : ℝ))) (nonneg : ∀ t ∈ Ioi (N : ℝ), 0 ≤ f t) : Summable (fun (n : ℕ) ↦ f n)` — with f : ℕ → ... wait, f n with n : ℕ — f must be ℕ → ℝ here? The anti is on ℝ... `f (↑n)` presumably with the cast. The statement: Summable (fun (n : ℕ) ↦ f n) where f : ℝ → ℝ and n coerced. So with N := 3: Summable (fun n : ℕ => f (↑n))... but I need Summable (fun n => log(n+4)/(n+4)²) = Summable (fun n => f (n+4)). From summable_nat_add_iff (N+1): Summable (fun n => f (n + N + 1))... The proof does `rw [← summable_nat_add_iff (N + 1)]` — so the full statement gives Summable (fun n ↦ f n) directly (the n + N + 1 shift is inside the proof). So `AntitoneOn.summable_of_integrableOn_Ioi anti int nonneg : Summable (fun n : ℕ => f ↑n)` — the WHOLE series from n = 0.\n\nFor my tail: Summable (fun n => f (n+4 : ℕ)): via summable_nat_add_iff: `summable_nat_add_iff (k : ℕ) : Summable (fun n => f (n + k)) ↔ Summable f` — so (summable_nat_add_iff 4).mpr hmm — Summable (fun n => f (n+4)) ↔ Summable f: mpr : Summable f → Summable (fun n => f (n+4)) ✓ — wait direction: the iff is Summable (shifted) ↔ Summable f, so .mpr : Summable f → Summable (shifted) ✓. Hmm but my h is stated as fun n => Real.log (n+4)/(n+4 : ℝ)^2 — the shift of (fun m => Real.log m/(m:ℝ)^2) by 4: (fun n => (fun m => log m/m²) (n+4)) = fun n => log (↑(n+4))/(↑(n+4))² — and my target has Real.log (n + 4) / (n + 4 : ℝ) ^ 2 — the casts: log (n+4) where n+4 is... I wrote `Real.log (n + 4) / (n + 4 : ℝ) ^ 2` — the (n+4) inside log: is it (n : ℝ) + 4 or ↑(n+4)? Real.log takes ℝ, so (n + 4) there elaborates as (n:ℝ)+4? or (↑(n+4))? Ambiguous — Lean will coerce n : ℕ → ℝ, so Real.log (↑n + 4). And (n + 4 : ℝ) is the ascribed form ↑(n+4)? (n + 4 : ℝ) with n : ℕ: the ascription forces the whole n+4 to be ℝ — elaborates as ↑n + 4 (the + at ℝ with coercion) or ↑(n+4) (the + at ℕ then cast) — Lean's default: (n + 4 : ℝ) with n : ℕ coerces n and does + in ℝ? The ascription (e : ℝ) makes e elaborate at ℝ: n + 4 at ℝ needs (n : ℝ) — coercion inserted: ↑n + 4. And ↑(n+4) vs ↑n+4 are defeq (Nat.cast_add) but not syntactic. My target in tsum_log_div_sq_tail_le was `∑' n : ℕ, Real.log (n + 4) / (n + 4 : ℝ) ^ 2` — the elaborated form... To avoid cast hell, let me state the tail consistently as `∑' n : ℕ, Real.log ((n : ℝ) + 4) / ((n : ℝ) + 4) ^ 2` — explicit ↑n + 4. Then the shift-lemma forms match cleanly via summable_nat_add_iff (4): Summable (fun n => g (n + 4)) where g m = log ↑m/↑m² — the shifted fun n => g (↑(n+4)) needs ↑(n+4) = ↑n + 4 — push_cast bridge.\n\nOK — this is getting into the weeds; let me just write the final §6 carefully and iterate on the build. First, repair the file: delete lines 222–431 (from the `/-- The integral of log...` docstring through the planning text), keeping dirichletTerm_norm_eq (lines 199–221) and the pieces 4/5 section (432+).\n\nActually the docstring at 201 starts dirichletTerm_norm_eq; the broken part starts at line 223 (the `/-- The integral of log x/x² over Ioi 3...` docstring). Let me delete from line 223 to line 431 (inclusive), then insert the new §6 content. Using sed for the deletion (precise line surgery), then Edit to insert.\n\nActually the instructions prefer dedicated tools over sed for edits, but for deleting a 209-line corrupt block, sed is the pragmatic choice... Alternatively, I can reconstruct the file: head -222 + new content + tail from 432. Let me do that with bash, writing the new middle via Write to a temp file. Actually — simplest correct approach: I'll read lines 1-222 (have them), write the new full file via Write (reconstructing the whole file: header + §1–§5 (unchanged, from my original Write + 4 Edits) + new §6 + §7 (pieces 4/5, unchanged from line 432+)). But the pieces-4/5 section content (lines 432–508) — I have it from my original Write. Let me reconstruct the whole file with Write. That's the safest. The file content I have fully: my original Write + the 4 edits (I know all their contents). Let me reconstruct.","signature":""},{"type":"text","text":"The file needs surgical repair (a bad edit pasted planning notes in). Reconstructing the full file cleanly with the factored §6."},{"type":"tool_use","id":"Bash_168","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1 && sed -n '432,508p' LiTightR.lean > /tmp/litightr_tail.txt && head -c 200 /tmp/litightr_tail.txt && sed -n '1,198p' LiTightR.lean > /tmp/litightr_head.txt && tail -5 /tmp/litightr_head.txt","description":"Extract the good head and tail of LiTightR.lean"},"extra_content":null}]