[{"type":"thinking","thinking":"More fixes:\n\n1. Line 1313: `Finset.sum_congr rfl h2` — the h2 as ∀ (i:ℕ) doesn't match the needed ∀ x ∈ range N. Fix: `Finset.sum_congr rfl (fun i _ => h2 i)`.\n\n2. Line 1328: HasSum's definition in this Mathlib is via `SummationFilter.unconditional` — not the range-partial-sums directly. `rw [HasSum]` unfolded to the unconditional filter form. Instead: use `hasSum_iff_tendsto_nat_of_summable`? The cleanest: `HasSum` on ℕ is defined via Tendsto of partial sums over `SummationFilter.unconditional`... Hmm — this Mathlib version changed HasSum's definition! The standard route: `tendsto_sum_nat_add`? Let me check what HasSum unfolds to and what lemmas give Tendsto-of-range-partial-sums → HasSum. There's `hasSum_iff_tendsto_nat_of_summable` — needs Summable first. Or `HasSum` def: HasSum f a := Tendsto (fun s => ∑ b ∈ s, f b) (SummationFilter.unconditional α).filter (𝓝 a)?? Wait the goal shows `Tendsto (fun s => ∑ b ∈ s, ...) (SummationFilter.unconditional ℕ).filter (𝓝 (1/4))`. Hmm — actually in current Mathlib: `HasSum f a := Tendsto (fun s : Finset α => ∑ b ∈ s, f b) atTop (𝓝 a)` where atTop on Finset — and for ℕ, `hasSum_iff_tendsto_nat_of_summable : HasSum f a ↔ Tendsto (fun n => ∑ i ∈ range n, f i) atTop (𝓝 a)` requires Summable f? no — `hasSum_iff_tendsto_nat_of_summable` is for summable f. There's `HasSum.tendsto_sum_nat`? And the converse: `HasSum.of_tendsto_sum_range`? hmm. The error shows HasSum unfolded to a `SummationFilter.unconditional` form — this is the NEW HasSum definition (mathlib changed it ~2024). The bridge: `HasSum` via `SummationFilter`... For a concrete proof, the cleanest: use `hasSum_iff_tendsto_nat_of_summable` — but needs Summable first — circular-ish: I can prove Summable FIRST via the telescope difference of convergent seq: (1/(k+4) − 1/(k+5)): each sequence converges, so summable via... hmm.\n\nLet me check what exists: search for \"SummationFilter\" / hasSum_iff_tendsto_nat in this Mathlib.\n\n3. Line 1357: farW_le_inv_mul_sub now needs 2 ≤ k+5 — fix the omega call: (by omega : 2 ≤ k + 5).\n\n4. Line 1364: div_mul_div_comm pattern mismatch — the goal is 1/((k+5)*(k+4)) = 1/(k+4)/(k+5): use field_simp: \n```\n_ = 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5) := by\n  field_simp\n```\n(1/(a*b) = 1/a/1·... — field_simp with ne conditions, or: rw [div_div, mul_comm]: 1/(k+4)/(k+5) = 1/((k+4)*(k+5)): div_div : a/b/c = a/(b*c) ✓ then mul_comm.)\n\n5. Line 1366: the abs_of_nonneg for 1/(k+4)/(k+5): `abs_of_nonneg (by positivity : (0:ℝ) ≤ 1/(k+4)/(k+5))` — the rw [Real.norm_eq_abs, abs_of_nonneg (farW_nonneg _), abs_of_nonneg ...] — the second abs_of_nonneg's argument form didn't match the goal atom: the goal after Real.norm_eq_abs has |farW (k+5)| and |1/(k+4)/(k+5)| — but ‖farW‖ is Real.norm which became |...| ✓ first abs_of_nonneg worked, second: the pattern |1/(↑k+4)/(↑k+5)| not found — because the term is written as `1 / (↑k + 4) / (↑k + 5)` = (1/(↑k+4))/(↑k+5) ✓ should match... the error says pattern not found in target `farW (k+5) ≤ 1/(↑k+4)/(↑k+5)` — there's NO abs around the RHS!! The `rw [Real.norm_eq_abs, abs_of_nonneg ...]` — Real.norm_eq_abs rewrote the LHS norm ‖farW‖ → |farW|, but the RHS of the ≤ in the goal has ‖1/...‖ — hmm the goal is `‖farW (k+5)‖ ≤ ‖1/(↑k+4)/(↑k+5)‖`?? of_norm_bounded gives ‖f k‖ ≤ ‖g k‖ with f = farW, g = telescope — Real.norm_eq_abs rewrites BOTH? rw rewrites all instances... The shown goal `farW (k+5) ≤ 1/(↑k+4)/(↑k+5)` has NO abs — meaning the two abs_of_nonneg rewrites already fired and cleared both norms, leaving the bare inequality which is exactly h2 — so the last `exact h2` should close it — but the error is that `abs_of_nonneg (by positivity : (0:ℝ) ≤ 1/(↑k+4)/(↑k+5))` — wait no, error 1366:4 is \"did not find |1/(↑k+4)/(↑k+5)|\" — the second abs_of_nonneg rewrite failed because... the first abs_of_nonneg (farW_nonneg _) rewrote |farW (k+5)| ✓. Then the second should find |1/(↑k+4)/(↑k+5)| — the goal display shows no abs — maybe the FIRST rw [Real.norm_eq_abs, abs...] — rw applies the list sequentially: Real.norm_eq_abs rewrote both ‖·‖ → |·|, then abs_of_nonneg (farW_nonneg _) rewrote |farW (k+5)| → farW (k+5) ✓, then abs_of_nonneg (0 ≤ 1/...) should rewrite |1/(↑k+4)/(↑k+5)| — the error says it wasn't found, and the displayed goal has no abs at all. Possibility: Real.norm_eq_abs didn't fire on the RHS because the RHS norm was already... OR the positivity proof-term's TYPE form mismatches: (0:ℝ) ≤ 1/(↑k+4)/(↑k+5) vs the abs argument |1/(↑k+4)/(↑k+5)| — abs_of_nonneg needs the inner expr to match EXACTLY. Hmm — the goal RHS: ‖1/(↑k+4)/(↑k+5)‖ — after Real.norm_eq_abs: |1/(↑k+4)/(↑k+5)| ✓ should match. Unless of_norm_bounded's goal is `‖farW (k+5)‖ ≤ ‖1/(↑k+4)/(↑k+5)‖` where the second ‖·‖ is Real.norm — Real.norm_eq_abs : ‖x‖ = |x| — rewrites ALL — including the LHS. Then abs_of_nonneg (farW_nonneg _) : |farW (k+5)| = farW (k+5) — rw rewrites it ✓. Then abs_of_nonneg (h : 0 ≤ 1/(↑k+4)/(↑k+5)) : |1/(↑k+4)/(↑k+5)| = 1/(↑k+4)/(↑k+5) — should fire... The error is at 1366:4 — let me look at the exact code lines to see the actual structure. The rw block was:\n```\n  rw [Real.norm_eq_abs, abs_of_nonneg (farW_nonneg _),\n    abs_of_nonneg (by positivity : (0 : ℝ) ≤ 1 / ((k : ℝ) + 4) / ((k : ℝ) + 5))]\n  exact h2\n```\nMaybe the issue: of_norm_bounded's goal uses ‖·‖ = Real.norm, and after Real.norm_eq_abs both become |·|, and abs_of_nonneg (farW_nonneg _) rewrote... the FIRST abs_of_nonneg's type is 0 ≤ farW (k+5)?? farW_nonneg _ : 0 ≤ farW ?k — the underscore unifies with k+5 ✓. Hmm the error message at 1366 says the PATTERN `|1 / (↑k + 4) / (↑k + 5)|` was not found in target `farW (k + 5) ≤ 1 / (↑k + 4) / (↑k + 5)` — the target already has no abs — meaning BOTH abs rewrites already happened?? No wait — if the second abs_of_nonneg FAILED, the state shown is AFTER the first succeeded. So |1/(↑k+4)/(↑k+5)| wasn't found — but the goal shows `1 / (↑k + 4) / (↑k + 5)` WITHOUT abs — contradiction: if the goal after first rewrite still had |1/...|, the error's goal display would show it. Unless the positivity proof term elaborated the division differently: (0:ℝ) ≤ 1/((k:ℝ)+4)/((k:ℝ)+5) — maybe positivity produced a proof about a DIFFERENTLY-associated expression?? No — proof irrelevance, only the TYPE matters for matching: 0 ≤ (1/(↑k+4)/(↑k+5)) — hmm.\n\nActually — maybe the issue is that `of_norm_bounded`'s goal is NOT ‖f‖ ≤ ‖g‖ but `‖f k‖ ≤ g' k` with g' already the norm... `Summable.of_norm_bounded (hg : Summable g) (h : ∀ i, ‖f i‖ ≤ g i)` — the RHS is g i NOT ‖g i‖!! (of_norm_bounded : Summable g → (∀ i, ‖f i‖ ≤ g i) → Summable f — with g real-valued, the condition is ‖f i‖ ≤ g i, no norm on g.) So the goal is ‖farW (k+5)‖ ≤ 1/(↑k+4)/(↑k+5) — only ONE norm. My rw: Real.norm_eq_abs (fires on LHS), abs_of_nonneg (farW_nonneg _) (fires ✓), then the third rewrite has no abs to find → error. Fix: drop the third rewrite.\n\n6. Line 1380/1383: farW_exact_four's `rw [farW]; norm_num` for the EQUALITY farW 1 = 1/√2 — error \"⊢ 0 ≤ 2\" and \"hsqrt2 : 1.414 ≤ √2 ⊢ ...\" — norm_num is trying to discharge `1 / (↑1 * √(1 + ↑1 ^ 2)) = 1 / √2` and failing into side goals. The issue: norm_num normalizes (↑1 : ℝ) → 1 and 1+1² → 2 but can't rewrite 1·√2·... — try `simp [farW]`? or explicit:\n```\nhave h2 : farW 1 = 1 / Real.sqrt 2 := by\n  rw [farW]\n  norm_num\n```\nfails. Do it manually:\n```\nhave h2 : farW 1 = 1 / Real.sqrt 2 := by\n  rw [farW]\n  rw [show ((1 : ℕ) : ℝ) = 1 by norm_num]\n  norm_num\n```\nHmm — the cast ((1:ℕ):ℝ): `Nat.cast_one`. Then goal: 1/(1·√(1+1^2)) = 1/√2: 1^2 = 1 via one_pow; 1+1 = 2; 1·√2 = √2 via one_mul. Chain: rw [Nat.cast_one, one_pow, one_mul]?? order: farW 1 = 1/((1:ℕ):ℝ... wait farW's def: 1/((k:ℝ)·√(1+(k:ℝ)^2)) with k = 1: ((1:ℕ):ℝ): rw [Nat.cast_one] : 1/(1·√(1+1^2)): rw [one_pow]: 1/(1·√(1+1)): rw [show (1:ℝ)+1 = 2 from rfl, one_mul] : 1/√2 ✓. So:\n```\nhave h2 : farW 1 = 1 / Real.sqrt 2 := by\n  rw [farW, Nat.cast_one, one_pow, show (1 : ℝ) + 1 = 2 from rfl, one_mul]\n```\nFor farW 2 = 1/(2·√5): rw [farW, Nat.cast_ofNat, ...]: ((2:ℕ):ℝ) = 2: Nat.cast_ofNat (simp lemma ✓); (2:ℝ)^2 = 4: norm_num; 1+4 = 5. \n```\nhave h3 : farW 2 = 1 / ((2 : ℝ) * Real.sqrt 5) := by\n  rw [farW, Nat.cast_ofNat]\n  norm_num\n```\n— after Nat.cast_ofNat: goal 1/(2·√(1+2^2)) = 1/(2·√5): norm_num: 2^2 = 4, 1+4 = 5 ✓ should close? norm_num can simplify 1+2^2 → 5 ✓ then both sides equal ✓. Risky but try. If not, explicit: rw [show (2:ℝ)^2 = 4 by norm_num, show (1:ℝ)+4 = 5 by norm_num]. Let me write all four with explicit show-rws:\n```\nhave h2 : farW 1 = 1 / Real.sqrt 2 := by\n  rw [farW, Nat.cast_one, one_pow, show (1 : ℝ) + 1 = 2 from rfl, one_mul]\nhave h3 : farW 2 = 1 / ((2 : ℝ) * Real.sqrt 5) := by\n  rw [farW, Nat.cast_ofNat, show (2 : ℝ) ^ 2 = 4 by norm_num, show (1 : ℝ) + 4 = 5 by norm_num]\nhave h4 : farW 3 = 1 / ((3 : ℝ) * Real.sqrt 10) := by\n  rw [farW, Nat.cast_ofNat, show (3 : ℝ) ^ 2 = 9 by norm_num, show (1 : ℝ) + 9 = 10 by norm_num]\nhave h5 : farW 4 = 1 / ((4 : ℝ) * Real.sqrt 17) := by\n  rw [farW, Nat.cast_ofNat, show (4 : ℝ) ^ 2 = 16 by norm_num, show (1 : ℝ) + 16 = 17 by norm_num]\n```\n\n7. The HasSum issue — biggest one. Let me find the right API in this Mathlib: the error shows `Tendsto (fun s => ∑ b ∈ s, f b) (SummationFilter.unconditional ℕ).filter (𝓝 a)` — hmm, that's odd; maybe this Mathlib version has HasSum defined via SummationFilter (very new). The natural bridge: `SummationFilter.tendsto_tsum`? or there should be `hasSum_iff_tendsto_nat_of_summable`? Let me search.","signature":""},{"type":"text","text":"Let me check this Mathlib's HasSum API (the SummationFilter form is new)."},{"type":"tool_use","id":"Bash_79","name":"Bash","input":{"command":"grep -rn \"hasSum_iff_tendsto_nat\\|tendsto_sum_nat\\|HasSum.*range\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/ --include=\"*.lean\" 2>/dev/null | grep \"theorem\\|lemma\" | head -10; echo ---; grep -rn \"def HasSum\\|theorem HasSum\\b\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Topology/Algebra/InfiniteSum/Defs.lean 2>/dev/null | head -5","description":"Find HasSum API in this Mathlib"},"extra_content":null}]