[{"type":"thinking","thinking":"Still failing. Line 3157 (farGLeftMaj_combo_summable's definition itself) times out at isDefEq at 200k — the TERM `(((tail14_inv_add_sq_summable.mul_left _).add ...).add ...).mul_left _` — the underscore metavars in a long chain of mul_left's — the elaborator must infer each coefficient — the .mul_left _ applications have types with the specific functions, and the final expected type is the combo-lambda — isDefEq between the built term's type and the ascription — Summable's function is inside the type... The isDefEq timeout suggests the elaborator is unfolding farGLeftMaj or the big lambdas pathologically.\n\nNew hypothesis: the real problem is that my combo-form function (a single lambda with three added scaled terms) has to be defeq-checked against the BUILT-UP Pi.add/Pi.mul forms, and isDefEq is slow on the giant real-literal polynomials (0.433*0.1942+0.886*0.0736+2.9 has ~6 operations, the log terms...). But 800k heartbeats?!\n\nAlternative: make the combo-summable proof STRUCTURAL instead of a term: \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  apply Summable.mul_left\n  apply Summable.add\n  apply Summable.add\n  · exact tail14_inv_add_sq_summable.mul_left _\n  · exact tail14_log_div_add_sq_summable.mul_left _\n  · exact tail14_loglog_div_add_sq_summable.mul_left _\n```\nThe apply-chain forces step-by-step unification — the final Summable (fun ℓ => f ℓ + g ℓ + h ℓ) vs the ascribed lambda — defeq by Pi.add_apply ✓ should be fast (small steps).\n\nHmm — but if the isDefEq of the final lambda against Pi.add-of-three is the problem, the apply chain hits the same wall. The whnf timeouts at 3214/3176 — those are DOWNSTREAM (series_le's hsplit etc.) — they may be fallout from combo_summable's failure (kernel unknown constant).\n\nLet me look at what's REALLY happening: use `set_option diagnostics true`? Can't easily iterate fast. Let me instead try compiling JUST the farGLeftMaj section in isolation to iterate faster... it's embedded mid-file. \n\nALTERNATIVE root-cause idea: `tail14_inv_add_sq_summable.mul_left _` — Summable.mul_left (r) : Summable f → Summable (fun x => r * f x) — the underscore for r — in the apply chain, the goal after `apply Summable.mul_left` : Summable (fun x => ?r * ...) — the ascription determines r ✓ fine.\n\nActually — WAIT. I bet the real issue is different: `farGLeftMaj` is a `def` with `if 14 ≤ ℓ then ... C0hat ... else 0` — and `farGLeftMaj_summable`'s proof `(summable_nat_add_iff (f := farGLeftMaj) 14).mp farGLeftMaj_shift_summable` — summable_nat_add_iff's statement: Summable (fun n => f (n + 14)) ↔ Summable f — .mp takes Summable (fun n => farGLeftMaj (n+14)) — farGLeftMaj_shift_summable has type Summable (fun ℓ => farGLeftMaj (ℓ + 14)) ✓ — fine.\n\nHmm, the timeout at 3157:0 — isDefEq — with the kernel-unknown-constant for combo_summable at 3166. So combo_summable's own elaboration timed out (isDefEq), leaving it undefined, cascading. The isDefEq is in the TERM proof — the final check of the built term against the ascribed Summable-type. Let me make combo_summable a tactic proof with the apply chain AND break the lambda into named pieces to help unification:\n\nActually — simpler robust approach: change the combo function to be DEFINITIONALLY built from the pieces: instead of writing the combo as a single big lambda, define the majorant as a def via the parts:\n\n```\ndef farGLeftMajShift (ℓ : ℕ) : ℝ :=\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))\n```\nand prove `Summable farGLeftMajShift` via: rw [farGLeftMajShift]; apply-chain. And farGLeftMaj_shift_le's conclusion: farGLeftMaj (ℓ+14) ≤ farGLeftMajShift ℓ. Then everywhere use farGLeftMajShift. The apply-chain for Summable farGLeftMajShift: after rw [farGLeftMajShift], apply Summable.mul_left/add/add — step unifications.\n\nIf the isDefEq STILL times out on the apply chain, the problem is deeper. Let me test quickly by editing just combo_summable to the apply-chain and recompiling — if it passes, the rest falls into place.\n\nAlso — 3214 and 3238 timeouts (isDefEq 800k, tactic execution 800k) at lines ~3214/3238 — those are in series_le (hdecomp's e1/e2 rfl-bridges probably). The rfl-bridges with the huge show-terms: `show (∑' x, big-lambda) = ∑' x, ((fun..) x + (fun..) x) from rfl` — the rfl requires isDefEq of the two big terms — heavy but should be fast (beta only)... The timeouts might all stem from combo_summable's kernel absence cascading (downstream references to it in hle's Summable.tsum_le_tsum args → isDefEq with a missing constant → hangs?). Fix combo first, then re-evaluate.","signature":""},{"type":"text","text":"Making combo_summable a structural apply-chain (and the same pattern wherever the term-level proof might choke)."},{"type":"tool_use","id":"Bash_198","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 = \"\"\"/-- The combo majorant is summable. -/\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))) :=\n  (((tail14_inv_add_sq_summable.mul_left _).add\n    (tail14_log_div_add_sq_summable.mul_left _)).add\n    (tail14_loglog_div_add_sq_summable.mul_left _)).mul_left _\"\"\"\nnew = \"\"\"/-- The combo majorant is summable. -/\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  apply Summable.mul_left\n  apply Summable.add _ (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 _)\"\"\"\nassert old in src\nsrc = src.replace(old, new)\n\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 -30","description":"Fix combo_summable as apply-chain, recompile","timeout":420000},"extra_content":null}]