[{"type":"thinking","thinking":"Issues:\n\n1. **Line 802 (summable_zpow_half_add_nat)**: the goal `1 / (↑m + 1 * (1/2))^(k+1) ≤ 2^(k+1) / (↑m+1)^(k+1)` — hmm — the goal after the zpow→one_div rewrites: `1 / (‖s‖...)` — wait this is the h2 block: my h2 : ((m:ℝ)+1)^(k+1) ≤ 2^(k+1)·((m+1/2))^(k+1) — and then the one_div application — the error shows the goal as `1/(↑m + 1·(1/2))^(k+1) ≤ 2^(k+1)/(↑m+1)^(k+1)` — hmm — the apply one_div_le_one_div_of_le's form: `?a ≤ ?b → 1/?b ≤ 1/?a` — so one_div_le_one_div_of_le (h : a ≤ b) : 1/b ≤ 1/a — ONE argument (the inequality)! My earlier usages passed (hpos) (hle) — two args — hmm — from the error: \"The full type of one_div_le_one_div_of_le ?m.227 is ?m.225 ≤ ?m.226 → 1/?m.226 ≤ 1/?m.225\" — so it takes ONE explicit arg (the inequality h : a ≤ b) — the positivity I passed as the first arg got consumed as the inequality?? — wait — my earlier successful usages: `one_div_le_one_div_of_le (by positivity) hpow` — hmm those compiled (LiSwaps/LiDigamma section 1-8)... — maybe there are TWO lemmas (one with the positivity arg + one without)? — the error shows the one-arg form. So `(by positivity)` got unified as the inequality → then hpow became... whatever — the earlier ones compiled — maybe the earlier `one_div_le_one_div_of_le (by positivity : (0:ℝ) < ...) hle` — hmm those compiled — so there's an overload with (0 < a) (h : a ≤ b)? — the current error's full type shows the ONE-ARG form — so the application `one_div_le_one_div_of_le (by positivity)` — the positivity-proof was taken AS the inequality — and failed to unify with the goal's 1/(...) ≤ 2^(k+1)/(...) — because the goal isn't a 1/· ≤ 1/· form (the RHS is 2^(k+1)/(...) — a div with numerator 2^(k+1)!). RIGHT — the goal after my rewrites is `1/(↑m+1·(1/2))^(k+1) ≤ 2^(k+1)/(↑m+1)^(k+1)` — NOT of the form 1/b ≤ 1/a — because I wrote the dominator family as 2^(k+1)/(m+1)^(k+1) (a div form). So my one_div_le_one_div_of_le application was wrong here — I need: 1/(m+1/2)^(k+1) ≤ 2^(k+1)/(m+1)^(k+1) ⟸ 2^(k+1)·(1/(m+1/2)^(k+1))·... — hmm: 1/A ≤ C/B ⟺ B ≤ C·A (positive): (m+1)^{k+1} ≤ 2^{k+1}·(m+1/2)^{k+1} — that's h2. So the step should be a div_le_div form: 1/A ≤ C/B with C = 2^{k+1}: via div_le_div_of_nonneg_left?? no — different numerators. Use `div_le_div (0 ≤ 1) (h2) (pos) (pos)`? — `div_le_div (ha : 0 ≤ a) (hab : a ≤ b) (hc : 0 < c) (hcd : c ≤ d)?: a/c ≤ b/d`? — hmm the modern form: `div_le_div (h0 : 0 ≤ b) (h1 : a ≤ b) (h2 : 0 < d) (h3 : c ≤ d)? : a / d ≤ b / c`?? — I never remember. Directly: 1/A ≤ C/B ⟺ (cross-multiplying) B ≤ C·A ⟺ 1 ≤ C·(A/B)... — let me use div_le_div_iff forms... — safest: use `div_le_div_iff (hb : 0 < b) (hd : 0 < d) : a/b ≤ c/d ↔ a·d ≤ c·b`:\ngoal 1/A ≤ C/B with A := (↑m+1/2)^(k+1), B := (↑m+1)^(k+1), C := 2^(k+1): div_le_div_iff (0 < A) (0 < B): 1/A ≤ C/B ↔ 1·B ≤ C·A — exactly h2 ✓:\n```lean\n  rw [div_le_div_iff (by positivity) (by positivity)]\n  ... goal: 1 * (↑m+1)^(k+1) ≤ 2^(k+1) * (↑m+1/2)^(k+1) — via one_mul then h2\n```\nHmm wait — but the goal's LHS is `1 / (↑m + 1 * (1/2))^(k+1)` — the `(↑m + 1 * (1/2))` — that's the norm form (m + 1/2) — displayed as ↑m + 1·(1/2)?? — the `1 * (1/2)` — push_cast normalization weirdness — whatever — the div_le_div_iff form handles it as an atom A. Let me restructure the end of summable_zpow_half_add_nat:\n```lean\n  rw [norm_zpow, h1]\n  rw [show (-(k : ℤ) - 1 : ℤ) = -(((k + 1 : ℕ) : ℤ)) by push_cast; ring, zpow_neg, zpow_natCast]\n  rw [div_eq_mul_inv, ← one_div, ← one_div]\n  rw [div_le_div_iff (by positivity : (0:ℝ) < ((m:ℝ) + 1/2)^(k+1)) (by positivity : (0:ℝ) < ((m:ℝ)+1)^(k+1))]\n  rw [one_mul]\n  -- goal: ((m:ℝ)+1)^(k+1) ≤ 2^(k+1)·(m+1/2)^(k+1)\n  rw [show (2 : ℝ) ^ (k + 1) * ((m : ℝ) + 1 / 2) ^ (k + 1) = (2 * ((m : ℝ) + 1 / 2)) ^ (k + 1) by rw [mul_pow]]\n  apply pow_le_pow_left₀ (by positivity : (0 : ℝ) ≤ (m : ℝ) + 1)\n  ring_nf\n  linarith [Nat.cast_nonneg (α := ℝ) m]\n```\nHmm — the `← one_div` rewrites: the goal after div_eq_mul_inv: ↑C * (X)⁻¹ ≤ (2^(k+1))/((↑m+1)^(k+1)) — wait — let me redo from the top. The goal at the start (of the of_norm_bounded side): ‖((1/2:ℂ)+↑m)^(−(k:ℤ)−1)‖ ≤ 2^(k+1)/((↑m+1)^(k+1)) — wait — the dominator in my refine was `fun m => (2^(k+1):ℝ)/((m:ℝ)+1)^(k+1)`?? — I passed `summable_div_add_one_pow (2 ^ (k + 1)) (k + 1) (by omega)` — that's Summable of fun m => 2^(k+1)/((m:ℝ)+1)^(k+1) ✓ and the goal: ‖series-term‖ ≤ (that family) m — the (that family) m is a lambda application (fun m => ...) m — beta issue AGAIN — the goal display shows the beta'd form ✓ (from the error: `1 / (↑m + 1 * (1/2))^(k+1) ≤ 2^(k+1)/(↑m+1)^(k+1)` — beta'd ✓). So after my norm_zpow/h1/exponent rewrites: the LHS norm became (↑m+1/2)^(−(k+1)) then zpow_neg/natCast then... my rewrites: `rw [norm_zpow, h1]` — norm_zpow: ‖X^(−(k+1):ℤ)‖ = ‖X‖^(−(k+1):ℤ) — hmm wait — norm_zpow : ‖x^q‖ = ‖x‖^q for q : ℤ ✓ — then h1 rewrites ‖(1/2)+↑m‖ → (↑m + 1/2) ✓ — then the exponent rewrite + zpow_neg + zpow_natCast: (↑m+1/2)^(−(k+1):ℤ) → ((↑m+1/2)^(k+1))⁻¹ — then `rw [div_eq_mul_inv, ← one_div, ← one_div]` — div_eq_mul_inv targets a ?a/?b — the goal at that point: ((↑m+1/2)^(k+1))⁻¹ ≤ 2^(k+1)/(↑m+1)^(k+1) — div_eq_mul_inv rewrites the FIRST /: the RHS's 2^(k+1)/(↑m+1)^(k+1) → 2^(k+1)·((↑m+1)^(k+1))⁻¹ ✓; then ← one_div ×2: turns the two ⁻¹ into 1/· ✓ then the goal: 1/((↑m+1/2)^(k+1)) ≤ 1·2^(k+1)/... hmm — I mis-sequenced again. Let me just carefully do:\n\ngoal (after norm_zpow, h1): (↑m + 1/2)^(−((k+1:ℕ):ℤ)) ≤ 2^(k+1)/((↑m+1)^(k+1)).\nrw [zpow_neg, zpow_natCast]: LHS → ((↑m+1/2)^((k+1:ℕ):ℤ))⁻¹ → wait zpow_neg : x^(−q) = (x^q)⁻¹ — with q := ((k+1:ℕ):ℤ) — LHS becomes ((↑m+1/2)^((k+1:ℕ):ℤ))⁻¹ then zpow_natCast: x^((n:ℕ):ℤ) = x^n — ((↑m+1/2)^(k+1))⁻¹ ✓.\nrw [div_eq_mul_inv] — the RHS 2^(k+1)/(...) → 2^(k+1)·((↑m+1)^(k+1))⁻¹.\nNow: ((↑m+1/2)^(k+1))⁻¹ ≤ 2^(k+1)·((↑m+1)^(k+1))⁻¹ — use `inv_le_iff`? or convert to one_div and use one_div... the forms have a coefficient on the RHS — the right lemma: `inv_le_iff...`? — let me just use field_simp-free calc:\n((↑m+1/2)^(k+1))⁻¹ ≤ 2^(k+1)·((↑m+1)^(k+1))⁻¹\n⟸ [mul by both positives]: (↑m+1)^(k+1) ≤ 2^(k+1)·(↑m+1/2)^(k+1) — this is h2.\nVia: `inv_le_iff_one_le_mul₀`? or... Let me use the one_div_le_one_div_of_le properly: 1/X ≤ 1/Y ⟸ Y ≤ X: but my RHS has the 2^(k+1) factor — convert: 2^(k+1)·((↑m+1)^(k+1))⁻¹ = 2^(k+1)·(1/(↑m+1)^(k+1)) = 2^(k+1)/(↑m+1)^(k+1) — hmm.\n\nSIMPLEST bulletproof: `rw [div_le_div_iff (0 < A) (0 < B)]` — goal 1/A ≤ C/B becomes 1·B ≤ C·A:\n- LHS: ((↑m+1/2)^(k+1))⁻¹ = 1/((↑m+1/2)^(k+1)) via one_div: rw [← one_div] on the LHS ⁻¹ — hmm — after zpow_neg+zpow_natCast the LHS is (X)^(−1)... — the ⁻¹ — rw [← one_div] turns X⁻¹ into 1/X — pattern ?a⁻¹ — matches ✓.\n- So goal: 1/((↑m+1/2)^(k+1)) ≤ 2^(k+1)·((↑m+1)^(k+1))⁻¹ — and the RHS has ⁻¹ — ← one_div on that too → 2^(k+1)·(1/(↑m+1)^(k+1)) — hmm — the RHS is 2^(k+1)·(1/X) — that's a product not a division — div_eq_mul_inv is a/b = a·b⁻¹ — reverse: rw [← div_eq_mul_inv] turns a·b⁻¹ into a/b ✓: RHS → 2^(k+1)/(↑m+1)^(k+1).\n- Then goal: 1/A ≤ C/B with A := (↑m+1/2)^(k+1), B := (↑m+1)^(k+1), C := 2^(k+1) — rw [div_le_div_iff (pos A) (pos B)] → 1·B ≤ C·A — one_mul → B ≤ C·A — the h2 content ✓.\n\nLet me rewrite the whole summable_zpow_half_add_nat with this careful sequence. And drop the h2-have/apply structure into this final form.\n\n2. **Line 863**: `tsum_eq_zero_add` unknown — the actual name: maybe `tsum_eq_zero_add` doesn't exist — it's `tsum_eq_zero_add`? hmm — let me think: the peel of a ℕ-tsum: `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n+1)` — hmm — maybe it's `tsum_eq_zero_add'`? or `series_eval...`? Let me search: there's `tsum_eq_zero_add` in Mathlib? — Actually I recall `tsum_eq_zero_add (hf : Summable f) : ∑' n, f n = f 0 + ∑' n, f (n + 1)` DOES exist (it's used for series reindexing). Let me grep.\n\n3. **Line 814**: \"unsolved goals\" with the context showing hA and hB... — the `rw [hA, hB]` — the error at 814:90 — hmm — 814 is hsplit's `rw [DinfC]`? — let me recount... — the error at 814:90 \"unsolved goals case e_a\" with the context showing hA/hB — hmm the context lists hA and hB as existing — so 814 is AFTER them?? no — the context shows hA exists — so the error is after hA's proof — at line 814?? that can't be right — the error positions shifted because the file changed. Let me recompile after fixing 1 and 2 and see what remains.","signature":""},{"type":"text","text":"Let me check `tsum_eq_zero_add`'s real name and fix the one_div sequence."},{"type":"tool_use","id":"Bash_482","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib\" && grep -rn \"theorem tsum_eq_zero_add\\|lemma tsum_eq_zero_add\" Mathlib/ --include=\"*.lean\" -r | head -3","description":"Find tsum_eq_zero_add"},"extra_content":null}]