[{"type":"thinking","thinking":"I see the problem at line 3163: `Summable.add` — checking its actual signature. From the error: \"Summable.add sorry (Summable.mul_left ...)\" — so Summable.add takes (hf)(hg) with the FUNCTIONS implicit — and `apply Summable.add` should leave the two proof-goals. The error \"could not unify Summable fun b => ?m.226 b + ... with the goal\" — the goal's lambda is `(A i + B i) + C i` and the result is `fun b => ?f b + C i` — unifying ?f := fun i => A i + B i — that failed — because... maybe Summable.add's result is `Summable (f + g)` — the Pi.add form — and the goal is the lambda — F1-unification of (Pi.add f g) with the lambda fails (needs ?f b + g b ≡ A i + B i + C i with the lambda structure — the elaborator should handle via first-order: f b = A i + B i ⟹ f := fun i => A i + B i? — no! (f + g) b = f b + g b — the goal's body: (A i + B i) + C i — matching f b + g b with g given (c2·C): g b = c2·C b ✓, f b = A i + B i ⟹ f := fun i => A i + B i ✓ — hmm — the b vs i binding: fun b => f b + c2·C b ≡ fun i => (A i + B i) + c2·C i — alpha-equiv + f-instantiation ✓ should work. The reported failure has ?m.226 UNRESOLVED in the displayed \"type of\" — meaning the elaborator gave up before solving ?f. This might be the isDefEq timeout path — the unifier was trying to be clever.\n\nNew approach — AVOID Summable.add's unification entirely: prove the combo summable via `Summable.of_norm_bounded` from an ALREADY-summable form built by the SAME lambda: no wait, that's circular.\n\nSIMPLEST BULLETPROOF: build the combo summable in three STEPS with named intermediate lemmas, each a direct `.mul_left`/`.add` at the term level with EXPLICIT function ascriptions matching the goal lambda EXACTLY:\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 hA : Summable (fun ℓ : ℕ =>\n      (0.433 * 0.1942 + 0.886 * 0.0736 + 2.9) * (1 / (((ℓ : ℝ) + 14) + 10) ^ 2)) :=\n    tail14_inv_add_sq_summable.mul_left _\n  have hB : Summable (fun ℓ : ℕ =>\n      0.433 * Real.log ((ℓ : ℝ) + 14) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    tail14_log_div_add_sq_summable.mul_left _\n  have hC : Summable (fun ℓ : ℕ =>\n      0.886 * Real.log (Real.log ((ℓ : ℝ) + 14)) / (((ℓ : ℝ) + 14) + 10) ^ 2) :=\n    tail14_loglog_div_add_sq_summable.mul_left _\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) := by\n    have h := hA.add hB\n    exact h\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) := by\n    have h := hAB.add hC\n    exact h\n  exact hABC.mul_left _\n```\nHere hA.add hB : Summable ((fun c0·A) + (fun c1·B)) : Summable (Pi.add) — and the ASCIBED type of hAB is the lambda `fun ℓ => c0·A ℓ + c1·B ℓ` — the `exact h` checks h's type Summable (Pi.add A B) against the ascription Summable (lambda A+B) — isDefEq: Pi.add A B ≡ fun ℓ => A ℓ + B ℓ — eta-expansion + Pi.add_apply — this SHOULD be fast defeq... The timeouts say it's NOT. WHY?!\n\nOH WAIT. I think I finally see it. `Summable.add` — let me check its actual signature in this Mathlib — maybe it's `Summable.add (hf : Summable f) (hg : Summable g) : Summable fun a => f a + g a` — result IS the lambda! Then hA.add hB : Summable (fun a => c0·A a + c1·B a) — EXACTLY the lambda ✓ — no Pi.add issue at all!! Then the earlier `apply Summable.add _ (hg)` — the apply's result Summable (fun a => ?f a + c2·C a) vs goal fun i => (A i + B i) + C i — the association: goal body is (A i + B i) + C i ✓ matches ?f a := A a + B a — should unify!! UNLESS the goal's association is A + (B + C)?? — my combo lambda: `X + Y + Z` parses `(X + Y) + Z` ✓ left-nested ✓.\n\nSo why does it fail?! The error at 3163 explicitly showed `Summable fun b => ?m.226 b + ?m.356·(...)` — ?m.226 is f — and it failed to unify with the goal. Hmm — maybe because Summable.add's signature in this Mathlib is `Summable.add {f g : α → M} (hf : Summable f) (hg : Summable g) : Summable (f + g)` with Pi.add, and the Pi.add unification against the lambda requires WHNF of Pi.add applied — and the elaborator's unification settings in `apply` (instances-implicit?) didn't reduce it. In that case `exact h` with the ascribed type would use the FULL defeq (isDefEq) which handles it — but the isDefEq TIMED OUT (800k)! On Pi.add?! That makes no sense unless something ELSE is slow.\n\nHmm — let me reconsider: maybe the slowdown is the LITERALS: `0.433 * 0.1942 + 0.886 * 736e-4 + 2.9` — the `736e-4` scientific-notation literal — its elaboration/whnf might be slow when repeated everywhere? And `0.433 * Real.log (↑i + 14)` — Real.log with cast chains. When isDefEq compares two such big types, it might unfold Real.log?? no.\n\nActually — WAIT. Let me look at the timeout at 3157 again (the isDefEq 200k at line 3157 which is `lemma farGLeftMaj_combo_summable`'s header/statement). Maybe the issue is the STATEMENT's elaboration itself — the function type with these terms — elaborating `Real.log (Real.log (↑i + 14))` — Real.log of Real.log — fine...\n\nYou know what — let me test the hypothesis that the LITERAL `736e-4` is the problem: replace all `736e-4` with `0.0736` (they're the same number — the pretty printer displayed 0.0736 as 736e-4). And `621e-4` → `0.0621`. If the scientific literals confuse the elaborator into expensive paths, this fixes it. Cheap to try.","signature":""},{"type":"text","text":"The error display shows literals as `736e-4`/`621e-4` (scientific notation). Let me replace them with plain decimals — possible elaboration pathology."},{"type":"tool_use","id":"Bash_201","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && python - <<'EOF'\np = 'NguyenRH1/LiTightR.lean'\nsrc = open(p, encoding='utf-8').read()\nn1 = src.count(\"736e-4\"); n2 = src.count(\"621e-4\")\nsrc = src.replace(\"736e-4\", \"0.0736\").replace(\"621e-4\", \"0.0621\")\nopen(p, 'w', encoding='utf-8').write(src)\nprint(\"replaced\", n1, n2)\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiTightR.lean 2>&1 | head -20","description":"Replace scientific literals and recompile","timeout":420000},"extra_content":null}]