[{"type":"thinking","thinking":"Two errors left:\n\n1. **Line 273** (integer case final linarith): the goal after rw [hnrm]: |−γ + ↑H| ≤ 20·log(2+‖↑n‖): with hreal : |−γ+H| ≤ γ + (1 + log(↑n−1)) and hlogn1 : log(↑n−1) ≤ log(2+‖w‖) — but ‖w‖ vs ‖↑n‖: the goal shows ‖↑n‖ (after rw [hwn], w became ↑n) while hlogn1 has ‖w‖ — hwn : w = ↑n but the hypothesis hlogn1 still mentions ‖w‖! The `rw [hwn, digamma_natCast_eq ...]` at the start rewrote w → ↑n in the GOAL but hlogn1 was proved later with ‖w‖... wait hlogn1's proof used hwnorm : ‖w‖ = ↑n — and hlogn1 : log(↑n−1) ≤ log(2+‖w‖). The goal has log(2+‖↑n‖). linarith treats log(2+‖w‖) and log(2+‖↑n‖) as DIFFERENT atoms. Fix: in hlogn1, end with the ‖↑n‖ form: after rw [hwnorm] the goal of hlogn1 is ↑n − 1 ≤ 2 + ↑n — hmm hlogn1 : log(↑n−1) ≤ log(2+‖w‖): the `apply Real.log_le_log hpos` leaves goal ↑n−1 ≤ 2+‖w‖: rw [hwnorm]: ↑n−1 ≤ 2+↑n ✓ linarith. So hlogn1 has ‖w‖. The goal needs ‖↑n‖. Add at the end: rw [hwn] at hlogn1? hwn : w = ↑n: rw [hwn] at hlogn1: rewrites w → ↑n ✓: hlogn1 : log(↑n−1) ≤ log(2+‖↑n‖) ✓. But careful: rw [hwn] earlier (line 232: `rw [hwn, digamma_natCast_eq n ...]`) rewrote the goal's w to ↑n — and `hreal`'s content doesn't mention w ✓. So just add `rw [hwn] at hlogn1` before the final linarith. But wait — the main-theorem goal after `rw [hwn, ...]`: shows ‖↑n‖ form — and hw_norm_ge still has ‖w‖... the final linarith uses hlog4 (which has ‖w‖)! hlog4 : 5/3 ≤ 19 log(2+‖w‖) — the goal has ‖↑n‖ — mismatch again. Simplest: don't rw the goal at all — instead convert hypotheses. Or: after `rw [hwn, digamma_natCast_eq n (by omega)]`, ALSO rw [hwn] at hw_norm_ge hlog4 — hmm cleaner: at the start of the integer case, `subst hwn`?? hwn : w = ↑n — subst needs a variable on one side: w is a free variable ✓ `subst hwn`?? subst works for h : x = t with x a local variable — w is a bound variable from the theorem's {w : ℂ} — can subst it? w is a variable ✓ but it appears in hwre hwim hypotheses — subst replaces everywhere ✓. But hwn : w = ↑n — subst w — wait subst eliminates w in favor of ↑n: `subst w`?? the tactic: `subst hwn` with hwn : w = ↑n: replaces w by ↑n everywhere ✓. But w came from {w : ℂ} implicit — subst'able ✓. Hmm — hwre : 2 ≤ (↑n:ℂ).re after subst — fine. Let me do: after hwn, add `subst hwn` — wait direction: subst hwn with hwn : w = ↑n: subst eliminates... `subst hwn` works when one side is a local variable not occurring in the other: w ✓ — replaces w := ↑n everywhere including hwre hwim hwn... but hwn itself is w = ↑n: after subst it becomes ↑n = ↑n and gets cleared ✓. But hold on — `rw [hwn, digamma_natCast_eq n (by omega)]` already rewrote the goal — if I subst instead, no rw needed: subst changes the goal's w to ↑n automatically: then `rw [digamma_natCast_eq n (by omega)]` — the goal has (↑n).digamma — the lemma's LHS pattern digamma ↑n: ✓.\n\nLet me restructure: after hwn, do `subst hwn` then `rw [digamma_natCast_eq n (by omega)]` and everything (hw_norm_ge, hlog4, hlogn1) uses ‖↑n‖ consistently. But — the hlogn1's proof used hwn : w = ↑n for hwnorm — after subst, hwn is gone: hwnorm : ‖(↑n:ℂ)‖ = ↑n: rw [RCLike.norm_natCast]?? goal ‖↑n‖ = ↑n: `RCLike.norm_natCast : ‖↑n‖ = ↑n`? hmm RCLike.norm_natCast (K) (n) : ‖(↑n : K)‖ = ↑n ✓: so hwnorm := by rw [RCLike.norm_natCast]... wait the goal would BE ‖↑n‖ = ↑n — rw [RCLike.norm_natCast] turns LHS into ↑n: rfl ✓. Simplify hlogn1's hwnorm to that. Hmm — this is getting complicated; SIMPLEST: keep hwn, and at the end do `rw [hwn] at hlogn1 hw_norm_ge hlog4`?? — hlog4's proof is already done (it's a have) — rw at haves is fine ✓. So before the final linarith: `rw [hwn] at hlogn1 hlog4` — wait also hw_norm_ge? not used in the final linarith directly (only via hlog4/hlogn1). The final linarith [hreal, hγ1, hlogn1, hlog4]: hreal : |..| ≤ γ + (1 + log(↑n−1)); hlogn1 : log(↑n−1) ≤ log(2+‖↑n‖) (after the rw); hlog4 : 5/3 ≤ 19 log(2+‖↑n‖) (after rw); goal: |..| ≤ 20 log(2+‖↑n‖). Chain: |..| ≤ γ + 1 + log(↑n−1) ≤ 1 + 1 + log(2+‖↑n‖) = 2 + log(2+‖↑n‖) and 2 ≤ ... hmm: need 2 + log X ≤ 20 log X ⟺ 2 ≤ 19 log X: log X = log(2+‖↑n‖) ≥ log 4 > 1: 19·1.38 = 26 ≥ 2 ✓ — but linarith needs log(2+‖↑n‖) ≥ (2/19)-ish — hlog4 gives 19·log ≥ 5/3: 5/3 ≥ 2? NO — 5/3 < 2!! hlog4 : 5/3 ≤ 19·log: that gives 19 log ≥ 1.67 — and I need 19 log ≥ 2. hlog4 was designed for the OTHER use (5/3 bound on the γ+1+... constant for the series case). For the integer case I need: γ + 1 + log(↑n−1) ≤ 20 log(2+‖↑n‖): γ + 1 ≤ 2 (γ ≤ 1 ✓ hγ1) and log(↑n−1) ≤ log(2+‖↑n‖) ✓: total ≤ 2 + log(2+‖↑n‖) ≤ 20 log(2+‖↑n‖) ⟺ 2 ≤ 19·log(2+‖↑n‖): log(2+‖↑n‖) ≥ log 4 ≥ 1: 19·1 = 19 ≥ 2 ✓ — but the linarith needs the explicit fact log(2+‖↑n‖) ≥ log 4 or ≥ 1: hlog4 gives 19·log ≥ 5/3 — 5/3 < 2 INSUFFICIENT for the needed 2 ≤ 19·log!! Wait: 19·log ≥ 5/3 ⟹ log ≥ 5/57 ≈ 0.0877; needed 2/19 ≈ 0.105 — 0.0877 < 0.105 INSUFFICIENT. So hlog4 (5/3) is too weak for the integer case's 2 ≤ 19 log. Hmm — actually: the integer case's constant is γ + 1 ≤ 2 exactly; the inequality needed: 2 ≤ 19·log(2+‖↑n‖): log(2+‖↑n‖) ≥ log 4 (‖↑n‖ = n ≥ 2): 19·log 4 ≈ 26.3 ≥ 2 ✓ TRUE. But hlog4 proves only 19·log ≥ 5/3. The linarith set [hreal, hγ1, hlogn1, hlog4] gives |..| ≤ 1 + 1 + log(2+‖↑n‖) (via hγ1: γ ≤ 1) and needs ≤ 20 log: ⟸ 2 + log ≤ 20 log ⟸ 2 ≤ 19 log — NOT implied by 19 log ≥ 5/3 (5/3 < 2). So I need a stronger hlog4-style fact: 2 ≤ 19·log(2+‖↑n‖) — from log ≥ log 4 > 1: 19·log 4 > 19 > 2 ✓. Add: have hlog2 : (2:ℝ) ≤ 19 * Real.log (2 + ‖↑n‖) := by nlinarith [the log4 > 1 fact]. Let me add a dedicated:\n```\n    have hlogbig : (2 : ℝ) ≤ 19 * Real.log (2 + ‖w‖) := by\n      -- log(2+‖w‖) ≥ log 4 > 1\n      have h1 : Real.log 4 ≤ Real.log (2 + ‖w‖) := Real.log_le_log (by norm_num) (by linarith [hw_norm_ge])\n      have h2 : (1 : ℝ) < Real.log 4 := ... (the hlog4's inner fact — refactor: extract `one_lt_log_four : 1 < Real.log 4` as a top-level helper to reuse)\n      nlinarith [h1, h2]\n```\nCleaner: extract a helper theorem `one_lt_log_four : (1:ℝ) < Real.log 4` at the top of §2 and use in both hlog4 and the integer case. And in the integer-case final: linarith [hreal, hγ1, hlogn1, hlogbig-with-‖↑n‖]... after the rw [hwn] at hlog4 — hmm. Let me restructure the integer case ending:\n```\n    rw [hwn] at hlog4 hw_norm_ge  -- now all in ‖↑n‖\n    have hlogbig : (2 : ℝ) ≤ 19 * Real.log (2 + ‖(n:ℂ)‖) := by nlinarith [hlog4]  -- 19·log ≥ 5/3... NO — 5/3 < 2!! \n```\nhlog4 gives 19 log ≥ 5/3 ≈ 1.67 — still < 2. The issue is hlog4's 5/3 was calibrated for the series case where the leftover constant is 5/3. For the integer case I need 2 ≤ 19 log — TRUE since 19 log 4 ≈ 26 ≫ 2 but NOT implied by 19 log ≥ 5/3. So hlogbig needs the log-4 fact directly. Refactor: helper `one_lt_log_four : (1:ℝ) < Real.log 4` (top-level, proved once) then:\n- hlog4 (series case): 5/3 ≤ 19·log(2+‖w‖) via log ≥ log 4 > 1: 19·1.38 ≥ 26 ≥ 5/3 ✓ nlinarith [h1, one_lt_log_four].\n- integer case: 2 ≤ 19·log(2+‖↑n‖) via 19·log 4 > 19 ≥ 2 ✓ nlinarith [h1', one_lt_log_four] with h1' : log 4 ≤ log(2+‖↑n‖).\n\nBoth nlinarith with [one_lt_log_four, the mono fact]: 19·log(2+‖↑n‖) ≥ 19·1 = 19 > 2: nlinarith: from log4 < log(2+‖↑n‖) and 1 < log4: log(2+‖↑n‖) > 1: 19·that > 19 ≥ 2 ✓ nlinarith handles.\n\n2. **Line 274** (hsplit unsolved goals — case hsplit AND case neg both shown): the hsplit's `simp only [] at h; rw [← h]` — the goal after: STILL the un-closed form — the error shows the goal `∑' ... = ∑ range N ... + ∑' (↑(n+N)...)` — so `rw [← h]` didn't close it. The h after simp only []: ∑ range N f + ∑' (fun n => f (n+N)) ... — the beta issue persists. Let me look at what h looks like: the error's goal has `(↑(n + N) + 1)` — and h's shifted form from sum_add_tsum_nat_add: f (n + N) with f the lambda — after simp only [] it should beta-reduce to the applied form `(↑(n+N) + 1)...` matching the goal — but it didn't close. Maybe simp only [] made no progress (would error \"no progress\"?) — no error shown for it, so it worked but the forms still differ. Difference: my goal's tail term: `((↑(n + N) + 1) * (A + ↑(n + N) + 1))⁻¹` — and h's: f (n+N) where f is the LAMBDA — after beta: `((↑(n+N):ℝ)+1) * (A + (↑(n+N):ℝ) + 1))⁻¹` hmm — vs goal's `(↑(n + N) + 1) * (A + ↑(n + N) + 1)` — wait the goal shows `↑(n + N)` — the Nat addition INSIDE the cast. And h's f (n + N): the application of f to (n + N : ℕ): after beta: (((↑(n+N) : ℝ) + 1) * (A + (↑(n+N):ℝ) + 1))⁻¹ — SAME as goal?! Then rw [← h] should close... unless the finite-sum part differs: goal's first part: `∑ n ∈ Finset.range N, ((↑n + 1) * (A + ↑n + 1))⁻¹` and h's: `∑ i ∈ Finset.range N, f i` with the LAMBDA unapplied — beta needed on h's side. simp only [] at h should beta-reduce. Did it error \"no progress\"? If simp only [] at h made NO progress it would ERROR — no such error shown — hmm, unless the error list got truncated (head -50 cut it). The two errors shown are 273 (linarith) and 274 (hsplit). Maybe simp only [] at h DID fail silently into the error at 274? The 274 error is \"unsolved goals case hsplit ... ⊢ ∑' ... = ∑ ... + ∑' ...\" — the FULL hsplit goal unsolved — meaning the by-block's last tactic left it. If simp only [] at h errored, that'd be a different message. So likely: simp only [] at h worked (beta-reduced) but the forms still mismatch subtly, OR rw [← h] failed to find the pattern... The `rw [← h]` with h : (finite sum) + (tsum shifted) = ∑' f: ← h rewrites occurrences of ∑' f (the RHS) with the LHS: my goal's LHS is ∑' n, ((↑n+1)*(A+↑n+1))⁻¹ — h's RHS: ∑' i, f i with f = the lambda: ∑' i, (fun n => ...) i — as a TERM this is tsum applied to the lambda — and my goal's LHS is tsum applied to (fun n => ((↑n+1)*(A+↑n+1))⁻¹) — SAME lambda?? h's f comes from hker_summ's type: Summable (fun n : ℕ => (((n:ℝ)+1)*(A+(n:ℝ)+1))⁻¹) — so f IS that lambda ✓ then h : ∑ range N (f i) + ∑' (f (i+N)) = ∑' f: with beta-redexes at the applications. simp only [] at h beta-reduces → h : ∑ i ∈ range N, ((↑i+1)*(A+↑i+1))⁻¹ + ∑' n, ((↑(n+N)+1)*(A+↑(n+N)+1))⁻¹ = ∑' n, ((↑n+1)*(A+↑n+1))⁻¹ — then rw [← h] at goal `∑' (my LHS) = ∑ (my head) + ∑' (my tail)`: ← h rewrites h's RHS (∑' n, kernel) → h's LHS: my goal's LHS IS h's RHS ✓ — after rw: goal = (h's LHS) = (my head + my tail): the head: h's ∑ i ∈ range N, ((↑i+1)...) vs my `∑ n ∈ Finset.range N, ((↑n+1)...)` — bound variable name i vs n — ALPHA-equivalent: are they syntactically equal for rw's rfl-close? Bound variable names don't matter for syntactic equality (de Bruijn) ✓. The tail: h's ∑' n, ((↑(n+N)+1)·...) vs my ∑' n, ((((n+N:ℕ):ℝ)+1)·...) — ↑(n+N) : the cast — hmm h's f (i+N) after beta: ((↑(i+N):ℝ)+1)·... — the cast is Nat.cast (↑(i+N)) ✓ same as mine ((n+N:ℕ):ℝ) ✓. So they SHOULD match... unless simp only [] didn't beta-reduce (maybe it needs `simp only []` to also unfold the... it does beta/eta ✓). \n\nHmm — wait, maybe the issue: `Summable.sum_add_tsum_nat_add` might have a DIFFERENT argument arrangement — maybe it's (h) (k) not (k) (h): from NatInt.lean line 233: `Multipliable.prod_mul_tprod_nat_add [T2Space G] {f} (k : ℕ) (h : Multipliable f)` — (k) THEN (h) — the to_additive: Summable.sum_add_tsum_nat_add (k : ℕ) (h : Summable f) — my call `hker_summ.sum_add_tsum_nat_add N`: dot notation: hker_summ fills the first explicit arg of matching type — h (Summable) — and N fills (k) ✓. OK.\n\nSo why didn't it close? Let me get the FULL error message for 274 — the \"case neg\" part was cut off. Let me recompile and look at the complete hsplit error. Actually — maybe the problem: `rw [← h]` rewrote the goal's LHS ∑' to (sum + tsum) — but there might be TWO tsums... my goal has TWO ∑' (LHS and the tail's). rw [← h] rewrites ALL occurrences of the pattern ∑' (kernel-lambda) — BOTH the LHS and the tail's ∑' (shifted form — different! the tail's is ↑(n+N) not ↑n — not the same pattern) — only the LHS matches ✓. Then the goal becomes h.LHS = (my head + my tail): if h's LHS (after beta) is syntactically my head + my tail, rw closes via rfl ✓. The error says UNSOLVED — so they differ. The difference must be in the finite-sum beta or the casts. Honestly — fastest: replace the whole hsplit proof with `exact (hker_summ.sum_add_tsum_nat_add N).symm` after converting via tsum_congr/sum_congr... Or: don't prove hsplit separately; inline. Let me just try variants at compile: first try `simp only []` on the GOAL too:\n```\n      have h := hker_summ.sum_add_tsum_nat_add N\n      simp only [] at h ⊢\n      rw [← h]\n```\nOr use convert: `convert h.symm using 2` (congruence depth 2 handles the per-term defeq):\nActually the truly robust: \n```\n      rw [show (∑' n : ℕ, ((((n + N : ℕ) : ℝ) + 1) * (A + ((n + N : ℕ) : ℝ) + 1))⁻¹)\n          = ∑' n : ℕ, (fun m : ℕ => (((m : ℝ) + 1) * (A + (m : ℝ) + 1))⁻¹) (n + N) from rfl]\n      rw [← hker_summ.sum_add_tsum_nat_add N]\n```\n— the show ... from rfl: my tail's lambda-applied form vs the redex: rfl if defeq ✓ (beta is defeq) — `from rfl` proves the equality of the two tsums (the functions are defeq) ✓. Then the goal's tail matches h's shifted form exactly and rw closes. Similarly the head's lambda might need the same treatment... the head: my ∑ n ∈ range N, ((↑n+1)·...)⁻¹ vs h's ∑ i ∈ range N, f i — f i redex: `(fun m => ...) i` vs applied — for the finite sum, h after simp only [] beta-reduces ✓ or add a similar show-rfl for the head:\n```\n      rw [show (∑ n ∈ Finset.range N, (((n : ℝ) + 1) * (A + (n : ℝ) + 1))⁻¹)\n          = ∑ n ∈ Finset.range N, (fun m : ℕ => (((m : ℝ) + 1) * (A + (m : ℝ) + 1))⁻¹) n from rfl]\n```\nHmm from rfl for the sums: the functions (fun n => body) vs (fun m => (lambda) m) — defeq ✓ the Finset.sum's are equal ✓ rfl works on defeq ✓.\n\nLet me write hsplit's proof with the two show-rfl bridges.","signature":""},{"type":"text","text":"Two errors left. Fixing both: the integer-case constant gap (5/3 < 2) and the hsplit beta-redex bridges."},{"type":"tool_use","id":"Read_320","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDigammaLeft.lean","offset":197,"limit":22},"extra_content":null}]