[{"type":"thinking","thinking":"The literals aren't the issue (0 replacements — the \"736e-4\" only appears in error output, the file has \"0.0736\").\n\nThe key error at 3163:21 — `Summable.add fun i => ...` — Summable.add is being applied to the FUNCTION — meaning Summable.add's first EXPLICIT arg is NOT the proof but... \"Application type mismatch: The argument fun i => ... has type (i : ℕ) → ?m.324 i but is expected to have type Summable ?m.226 ?m.228\" — so Summable.add expects its first arg to be a Summable PROOF. So Summable.add (hf : Summable f) (hg : Summable g) : Summable (f + g)? — and my `apply Summable.add (fun i => ...)` wrongly passed a function. That was MY error from the last patch (I wrote `apply Summable.add (fun i => ...) (...)`).\n\nLet me check the exact Summable.add signature — it's likely `Summable.add {f g : ι → M} (hf : Summable f) (hg : Summable g) : Summable fun x => f x + g x` (from the error's expected Summable (fun b => ?f b + ?g·C b) — result IS a lambda ✓). So the WORKING pattern is: hA.add hB — term-level. And the goal-vs-result unification: Summable (fun x => c0·A x + c1·B x) — the goal lambda — this SHOULD unify syntactically (both lambdas with + bodies) ✓.\n\nSo the earlier `exact (h1.mul_left _).add (h2.mul_left _)` failure — h1.mul_left _ : Summable (fun x => c0·A x) — wait — does Summable.mul_left return `Summable (fun x => r * f x)` or `Summable (r * f)` (Pi.mul)? From the error display: `Summable.mul_left ?m.356 tail14_loglog_div_add_sq_summable` has type `Summable fun b => ?m.356 * (Real.log (Real.log (↑b + 14)) / (↑b + 14 + 10) ^ 2)` — a LAMBDA with * inside ✓. So hA.add hB : Summable (fun x => c0·A x + c1·B x) ✓ lambda ✓. Then hAB.add hC : Summable (fun x => (c0·A x + c1·B x) + c2·C x) ✓ matches the goal lambda ✓. So:\n\n```\nlemma farGLeftMaj_combo_summable : Summable (fun ℓ : ℕ =>\n    4.5 * ((0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)\n        + (0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2)\n        + (0.886 * Real.log (Real.log ((ℓ : ℝ) + 14)) / (((ℓ : ℝ) + 14) + 10) ^ 2))) := by\n  have hAB : Summable (fun ℓ : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)\n      + 0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    (tail14_inv_add_sq_summable.mul_left _).add (tail14_log_div_add_sq_summable.mul_left _)\n  have hABC : Summable (fun ℓ : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)\n      + 0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2\n      + 0.886 * Real.log (Real.log ((ℓ : ℝ) + 14)) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    hAB.add (tail14_loglog_div_add_sq_summable.mul_left _)\n  exact hABC.mul_left _\n```\nThe `hABC.mul_left _` : Summable (fun ℓ => 4.5 * (hABC-body ℓ)) — the goal: 4.5 * (A+B+C) — the mul_left result: fun ℓ => 4.5 * ((c0·A + c1·B) + c2·C) — the goal's body: 4.5 * ((A+B) + C)... wait — my goal lambda is `4.5 * (c0·A + c1·B + c2·C)` — hmm — the c0·A term in the goal: `(0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / ...)` — and hABC's body has THE SAME ✓. And the mul_left gives fun ℓ => 4.5 * hABC-body ℓ ✓ syntactic match ✓.\n\nThen the timeouts at 3217/3241/3179 were downstream of combo being missing. Let me apply this and recompile.","signature":""},{"type":"text","text":"Found it — `Summable.add` takes proofs, not functions (my last patch's mistake). Fixing with the term-level chain."},{"type":"tool_use","id":"Bash_202","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && python - <<'EOF'\np = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\n\nold = \"\"\"  apply Summable.mul_left (4.5 : ℝ)\n  apply Summable.add (fun i : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((i : ℝ) + 14) + 10) ^ 2)\n      + 0.433 * Real.log ((i : ℝ) + 14) / (((i : ℝ) + 14) + 10) ^ 2)\n    (tail14_loglog_div_add_sq_summable.mul_left _)\n  exact (tail14_inv_add_sq_summable.mul_left _).add (tail14_log_div_add_sq_summable.mul_left _)\"\"\"\nnew = \"\"\"  have hAB : Summable (fun ℓ : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)\n      + 0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    (tail14_inv_add_sq_summable.mul_left _).add (tail14_log_div_add_sq_summable.mul_left _)\n  have hABC : Summable (fun ℓ : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)\n      + 0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2\n      + 0.886 * Real.log (Real.log ((ℓ : ℝ) + 14)) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    hAB.add (tail14_loglog_div_add_sq_summable.mul_left _)\n  exact hABC.mul_left _\"\"\"\nassert old in src\nsrc = src.replace(old, new)\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"done\")\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiTightR.lean 2>&1 | head -25","description":"Fix combo summable with term-level chain, recompile","timeout":420000},"extra_content":null}]