[{"type":"thinking","thinking":"Variant 1: `(testA.mul_left _).add (testB.mul_left _)` — \"don't know how to synthesize implicit argument g\" — so Summable.add's signature here is `Summable.add {f g : ι → M} {S : SummationFilter} [AddCommMonoid M] ... (hf : Summable f S?) : Summable (fun x => f x + g x)`?? — the error shows `@Summable.add ℝ ℕ ... (fun i => ?m.211 * ...) (fun i => ?m.227 * ...) (SummationFilter.unconditional ℕ) ...` — the ARGS of Summable.add here: {f}, {g}, (SummationFilter), (instance), (hf) — and it says \"don't know how to synthesize implicit argument g\" — Summable.add's hg (the g summability) is an IMPLICIT ARG?! No wait — the displayed form shows only ONE proof arg (Summable.mul_left ?m.211 testA) — so Summable.add took hf explicitly and hg is... — the signature must be `Summable.add (hf : Summable f) (hg : Summable g) : Summable fun x => f x + g x` — and .add (testC.mul_left _) supplied hg — but the error says g (the FUNCTION) can't be synthesized — because hg's type has ?m.227·B — the g FUNCTION has a metavar coefficient — and then the final goal-check needs to unify... The \"don't know how to synthesize implicit argument g\" — g is implicit and hg doesn't determine it (because hg = testC.mul_left _ with ?c metavar!!). THAT's the issue — the underscore in testC.mul_left _ leaves ?c unresolved, and hg's type is Summable (fun i => ?c·C i) — g := fun i => ?c·C i — the unification with the GOAL happens at the `exact` — and the error appears at hAB.add's ELABORATION (line 15, the exact hAB.add (...) line): the g-function has a metavar and Summable.add's implicit g must be determined FROM hg's type — it IS determined (as ?c·C-fun) — but the \"synthesize implicit argument g\" error says it can't — because ?c is still a metavar at that point!\n\nThe FIX: give the coefficients explicitly in the mul_left calls: `(testC.mul_left 0.886)`. Then g is fully determined and the final goal-unification works. Let me verify.","signature":""},{"type":"text","text":"The underscores in `mul_left _` leave metavariables that break downstream unification. Testing explicit coefficients."},{"type":"tool_use","id":"Bash_207","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1 && cat > /tmp/test_lam4.lean <<'EOF'\nimport Mathlib\nopen Complex Filter Topology MeasureTheory\nnoncomputable section\nnamespace TestNS\n\nlemma testA : Summable (fun ℓ : ℕ => (1:ℝ) / (((ℓ:ℝ)+14)+10)^2) := by sorry\nlemma testB : Summable (fun ℓ : ℕ => Real.log ((ℓ:ℝ)+14) / (((ℓ:ℝ)+14)+10)^2) := by sorry\nlemma testC : Summable (fun ℓ : ℕ => Real.log (Real.log ((ℓ:ℝ)+14)) / (((ℓ:ℝ)+14)+10)^2) := by sorry\n\nlemma test4 : 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    (testA.mul_left (0.433*0.1942+0.886*0.0736+2.9)).add (testB.mul_left 0.433)\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 (testC.mul_left 0.886)\n  exact hABC.mul_left 4.5\n\nend TestNS\nEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && time lake env lean /tmp/test_lam4.lean 2>&1 | head -8","description":"Test explicit coefficients in the chain","timeout":300000},"extra_content":null}]