[{"type":"thinking","thinking":"A cascade of issues. Let me fix them:\n\n1. **Line 102 (hh)**: `Rat.cast_sum` didn't match `↑(∑ ...)` — the cast of a sum. The issue: `Rat.cast_sum` might have a different name or the cast is a different mechanism. The term: `↑(∑ i ∈ Finset.range n, (↑(i + 1))⁻¹)` — the ↑ is `Rat.cast`? — for Rat.cast into ℝ: the lemma is `Rat.cast_sum : (↑(∑ i ∈ s, f i) : ℝ) = ∑ i ∈ s, (↑(f i) : ℝ)`? — the error says pattern `↑(∑ x ∈ ?s, ?f x)` not found — the rw's lemma LHS didn't match. Maybe the current name is different (map_sum for the ring hom!). Use `map_sum (Rat.castHom ℝ)?`... — hmm — the cleanest: `push_cast` should push the cast through the sum: \n```lean\n  have hh : (harmonic n : ℝ) = ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) := by\n    rw [harmonic]\n    push_cast\n    ...\n```\npush_cast on ↑(∑ ...) → ∑ ↑(terms) — via the cast lemmas (push_cast knows Rat.cast_sum etc.) ✓. Then the per-term casts: ↑((↑(i+1):ℚ)⁻¹) → (↑(i+1):ℝ)⁻¹ → ((↑i:ℝ)+1)⁻¹ — push_cast normalizes to that; the RHS has 1/(↑i+1) — after push_cast the goal's LHS is a sum — hmm push_cast doesn't produce sum_congr form... push_cast on the GOAL ↑(∑ ...) = ∑ (1/(↑i+1)): LHS becomes ∑ i, (↑(i+1):ℝ)⁻¹ — then need per-term (↑(i+1):ℝ)⁻¹ = 1/(↑i+1) — inv_eq_one_div — simp: `inv_eq_one_div`? or one_div — so:\n```lean\n    rw [harmonic]\n    push_cast\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [one_div]\n```\nHmm — after push_cast the goal: ∑ i, (↑(i + 1) : ℝ)⁻¹ = ∑ i, 1/(↑i + 1) — sum_congr + one_div: 1/(↑i+1) → (↑i+1)⁻¹ ✓ then rfl ✓. Hmm — but does push_cast rewrite the outer cast of the sum into a sum of casts? push_cast uses `map_sum`-style lemmas ✓ (Rat.cast is a ring hom — push_cast handles) ✓. And ↑(i+1):ℚ → ... — the inner cast ↑(i+1) with i : ℕ — the term is (↑(i + 1) : ℚ)⁻¹ with ↑(i+1) : ℚ via Nat.cast (i+1)?? — hmm the harmonic def: ∑ i ∈ range n, (↑(i + 1))⁻¹?? — from the error display: `(↑(i + 1))⁻¹` — so ↑(i+1) with i+1 computed in ℕ then cast to ℚ ✓. push_cast: ↑(↑(i+1) : ℚ)⁻¹ : ℝ — push_cast maps to ((↑(i+1) : ℝ))⁻¹ then ((↑i + 1 : ℝ))⁻¹ ✓ (Nat.cast_add, Nat.cast_one are push_cast lemmas ✓). So after push_cast: LHS = ∑ i, ((↑i:ℝ)+1)⁻¹ ✓ then sum_congr + one_div ✓.\n\n2. **Line 111 (h2)**: after rw [Finset.sum_range_succ'] — the goal became `∑ k ∈ range n, 1/(x+↑(k+1)) + 1/(x+↑0) = 1/x + ∑ i ∈ range n, 1/(x+↑i+1)` — and my `rw [Finset.sum_congr rfl (fun i _ => e i)]` failed: the pattern `∑ i ∈ ?s, 1/(x+(↑i+1))` — hmm the goal's first sum is `∑ k ∈ range n, 1/(x+↑(k+1))` — the pattern `∑ i ∈ ?s, 1/(x+(↑i+1))` should match with i := k... — Finset.sum_congr rfl (h) : ∑ i ∈ s, f i = ∑ i ∈ s, g i — as a REWRITE rule — rw with it rewrites the LHS-pattern sum to the RHS — the issue: rw needs a universally-quantified equality — Finset.sum_congr rfl (fun i _ => e i) : ∑ k ∈ range n, 1/(x+↑(k+1)) = ∑ k ∈ range n, 1/(x+↑k+1) — wait — e i : 1/(x+(↑i+1)) = 1/(x+↑i+1) — the sum_congr with f k := 1/(x+↑(k+1)) and g k := 1/(x+↑k+1) — the term-level form: ∑ k ∈ range n, (fun k => 1/(x+↑(k+1))) k vs the goal's `∑ k ∈ range n, 1/(x+↑(k+1))` — the sum in the goal has the lambda beta-reduced — sum_congr's statement produces ∑ f = ∑ g with f g as lambdas — the match between `∑ k, (fun k => ...) k` and `∑ k, 1/(x+↑(k+1))` — Lean's rw matching up to reducible defeq (beta) — should match... The error says pattern `∑ i ∈ ?m, 1/(x+(↑i+1))` not found — hmm — maybe the summand in the goal is displayed as `1 / (x + ↑(k + 1))` (the ↑(k+1) — cast of the Nat-sum k+1) — and my pattern `1 / (x + (↑i + 1))` — (↑i + 1) is ↑(i)+1 — DIFFERENT from ↑(k+1)!! — the goal has ↑(k+1) (cast of the sum) and my e's LHS pattern has (↑i + 1) (sum of cast and literal) — NOT syntactically equal! e's LHS: `1 / (x + (↑i + 1))` — I wrote `1 / (x + (i + 1 : ℝ))` — `(i + 1 : ℝ)` with i : ℕ — that's (↑i + 1 : ℝ)? — `(i + 1 : ℝ)` elaborates i at ℝ: ↑i, then + 1 ✓ so e's LHS has (↑i + 1). But the goal (from sum_range_succ's f (k+1)) has ↑(k+1) — the CAST of k+1. So mismatch. Fix: e's LHS should be `1 / (x + ((k + 1 : ℕ) : ℝ))`:\n```lean\n    have e : ∀ i : ℕ, 1 / (x + ((i + 1 : ℕ) : ℝ)) = 1 / (x + i + 1 : ℝ) := fun i => by\n      push_cast\n      ring_nf\n```\nThe push_cast turns ↑(i+1) into ↑i + 1 ✓ then both sides 1/(x+↑i+1) — rfl-ish — ring_nf might close; or after push_cast the two sides are syntactically equal → the goal closes by rfl automatically? push_cast leaves `1/(x+(↑i+1)) = 1/(x+↑i+1)` — association — ring_nf ✓.\n\n3. **Line 119-120**: the h3 block — I mis-structured. Current code (from my edit):\n```lean\n  have h3 : ∑ i ∈ Finset.range n, (1 / ((i : ℝ) + 1) - 1 / (x + i + 1)) =\n      ∑ i ∈ Finset.range n, dpsiTermR x i := by\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [dpsiTermR]\n    ring\n  rw [← Finset.sum_sub_distrib] at h3\n  rw [← h3]\n  ring\n```\nThe `rw [dpsiTermR]; ring` — line 119's \"No goals\" — the rw [dpsiTermR] closed the goal by rfl, then ring errored ✓ remove ring. Then `rw [← Finset.sum_sub_distrib] at h3` — h3's LHS is the sub-SUM — ← sum_sub_distrib turns Σ f − Σ g INTO Σ (f − g) — but h3's LHS is ALREADY Σ (f − g) form — I want the OTHER direction... wait: h3 : Σ (f − g) = Σ dpsiTermR. I want to rewrite the main goal — the main goal (from the error): `Real.log ↑n − (1/x + Σ 1/(x+↑i+1)) = Real.log ↑n − Σ 1/(↑i+1) + (−1/x + Σ dpsiTermR x m)`. So the pieces: A := Σ 1/(↑i+1) (inside RHS's first sum), B := Σ 1/(x+↑i+1) (LHS), C := Σ dpsiTermR (RHS). h3 : Σ(f−g) = C where Σ(f−g) = A − B after sum_sub_distrib... The clean path: prove h3' : C = A − B and rw [h3'] on the main goal's C, then ring:\n```lean\n  have h3 : ∑ i ∈ Finset.range n, dpsiTermR x i =\n      ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) - ∑ i ∈ Finset.range n, 1 / (x + i + 1) := by\n    rw [← Finset.sum_sub_distrib]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [dpsiTermR]\n  rw [h3]\n  ring\n```\nh3's proof: goal Σ dpsiTermR = Σ (1/..) − Σ (1/..) — rw [← Finset.sum_sub_distrib] turns RHS into Σ (1/.. − 1/..) ✓ then sum_congr + rw [dpsiTermR] — per-term goal dpsiTermR x i = 1/(↑i+1) − 1/(x+↑i+1) — rw [dpsiTermR] unfolds to that ✓ rfl-closes by rw ✓.\nThen main goal after rw [h3] — hmm wait — the main goal at that point: after rw [hh] and rw [h2]:\nLHS: Real.log ↑n − (1/x + B)  [h2 rewrote the big sum]\nRHS: (Real.log ↑n − A) + (−1/x + C)  [hh rewrote harmonic]\nrw [h3]: C → A − B ✓ then ring: log n − (1/x + B) = log n − A + (−1/x + (A − B)) ✓ ring closes (A B atoms; 1/x an atom to ring — division by x — ring treats 1/x as x⁻¹ atom ✓ consistent both sides ✓).\nOrder: I need rw [hh, h2, h3] in the right sequence — let me restructure dpsiSeqR_eq:\n```lean\nlemma dpsiSeqR_eq (x : ℝ) (n : ℕ) :\n    dpsiSeqR x n = (Real.log n - (harmonic n : ℝ)) +\n      (-1 / x + ∑ m ∈ Finset.range n, dpsiTermR x m) := by\n  rw [dpsiSeqR]\n  have hh : (harmonic n : ℝ) = ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) := by\n    rw [harmonic]\n    push_cast\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [one_div]\n  have h2 : ∑ m ∈ Finset.range (n + 1), 1 / (x + (m : ℝ)) =\n      1 / x + ∑ i ∈ Finset.range n, 1 / (x + i + 1) := by\n    rw [Finset.sum_range_succ']\n    have e : ∀ i : ℕ, 1 / (x + ((i + 1 : ℕ) : ℝ)) = 1 / (x + i + 1 : ℝ) := fun i => by\n      push_cast\n      ring_nf\n    rw [Finset.sum_congr rfl (fun i _ => e i)]\n    have e0 : (fun m => 1 / (x + (m : ℝ))) 0 = 1 / x := by simp\n    rw [e0]  -- hmm the f 0 term\n    ring\n  have h3 : ∑ i ∈ Finset.range n, dpsiTermR x i =\n      ∑ i ∈ Finset.range n, 1 / ((i : ℝ) + 1) - ∑ i ∈ Finset.range n, 1 / (x + i + 1) := by\n    rw [← Finset.sum_sub_distrib]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [dpsiTermR]\n  rw [hh, h2, h3]\n  ring\n```\nWait — order of rewrites in the main goal: after rw [dpsiSeqR], the goal is:\n`Real.log ↑n − Σ_{range(n+1)} 1/(x+↑m) = (Real.log ↑n − ↑harmonic) + (−1/x + Σ_{range n} dpsiTermR x m)`.\nrw [hh]: rewrites ↑harmonic (RHS) → Σ 1/(↑i+1) ✓.\nrw [h2]: rewrites Σ_{range(n+1)} 1/(x+↑m) (LHS) → 1/x + Σ_{range n} 1/(x+↑i+1) ✓.\nrw [h3]: rewrites Σ_{range n} dpsiTermR x m (RHS) → A − B ✓.\nring ✓.\n\nHmm — the h2's e0: after rw [Finset.sum_congr rfl e], the goal: (∑_{i∈range n} 1/(x+↑i+1)) + (fun m => 1/(x+↑m)) 0 = 1/x + Σ... — the f 0 term as the application — e0 : (fun m => 1/(x+↑m)) 0 = 1/x by simp — simp computes: ↑0 = 0, x + 0 = x... — hmm — (fun m => ...) 0 beta → 1/(x + ↑0) → simp [Nat.cast_zero, add_zero] → 1/x ✓ simp should do it. Then ring: (S) + 1/x = 1/x + S ✓ — where S := Σ_{range n} 1/(x+↑i+1) — but wait — the two sums: LHS's (after congr) ∑_{i∈range n} 1/(x + ↑i + 1) and RHS's ∑_{i∈range n} 1/(x + i + 1) — the `x + i + 1` in the h2 statement's RHS: I wrote `1 / (x + i + 1)` with i : ℕ — elaborates as 1/(x + ↑i + 1) ✓ same ✓.\n\nAlso — `harmonic` — need to confirm: is it `harmonic n : ℚ` and does `rw [harmonic]` unfold? harmonic is a def — rw [harmonic] uses the equation lemma ✓ should work — hmm — actually for `def harmonic (n : ℕ) : ℚ := ...` the equation lemma `harmonic.eq_def`? or `harmonic n` unfolding — rw [harmonic] might fail for a def with an argument — the auto-generated equation: harmonic n = ∑ ... ✓ rw [harmonic] should rewrite `harmonic n` ✓ — if it fails use `simp only [harmonic]` or `show`.\n\nWait — one more thing about harmonic: is the def `harmonic n := ∑ i ∈ Finset.range n, (↑i + 1)⁻¹` (i cast then +1) or (↑(i+1))⁻¹? From the error display earlier: `(↑(i + 1))⁻¹` — so ↑(i+1) — the Nat-sum cast. So after push_cast: (↑(i+1):ℝ)⁻¹ → (↑i+1)⁻¹ ✓.\n\n4. **Line 129 (exists_sInf_mem)**: `hKc.exists_sInf_mem` — hKc : IsCompact K — but the error shows hKc's type as `?m ≤ 𝓟 K → ∃ x ∈ K, ClusterPt x ?m` — that's IsClusterPt-related — my obtain destructured WRONG: I wrote `obtain ⟨c, hcK, hc0⟩ := hKc.exists_sInf_mem (...)` — but the error says exists_sInf_mem isn't a field of IsCompact — it IS `IsCompact.exists_sInf_mem`? — maybe the name is different: for IsCompact K in ℝ: the min-attainment: `IsCompact.exists_sInf_mem : IsCompact s → s.Nonempty → sInf s ∈ s`? — hmm the error says \"environment does not contain Function.exists_sInf_mem\" — the dot-notation resolved weirdly because the ⟨...⟩ obtain pattern mismatched first. The correct name: `IsCompact.exists_sInf_mem`? or `IsCompact.sInf_mem`? — In Mathlib: for a compact nonempty set of reals, sInf K ∈ K: I believe `IsCompact.exists_sInf_mem (hKc : IsCompact s) (hne : s.Nonempty) : sInf s ∈ s`? — hmm — it's `IsCompact.exists_sInf_mem`? Let me check: candidates — `IsCompact.exists_sInf_mem`, `IsCompact.sInf_mem`, `IsCompact.exists_isMinOn`... The reliable one: `IsCompact.exists_isMinOn : IsCompact s → s.Nonempty → Continuous f → ∃ x ∈ s, IsMinOn f s x` — for f = id: gives c ∈ K with ∀ y ∈ K, c ≤ y ✓ that's cleaner anyway:\n```lean\n  obtain ⟨c, hcK, hcle⟩ := hKc.exists_isMinOn hKne (continuous_id.continuousOn)\n  -- hcle : IsMinOn id K c — i.e. ∀ y ∈ K, id c ≤ id y\n```\nIsMinOn id K c : ∀ y ∈ K, id c ≤ id y — IsMinOn f s x := ∀ y ∈ s, f x ≤ f y ✓ so hcle y hy : id c ≤ id y → c ≤ y ✓.\nSimilarly sSup: `IsCompact.exists_isMaxOn`.\nHmm — but I need K.Nonempty for those — from hK : K ⊆ Ioi 0 and... K could be EMPTY! If K = ∅ the statement is vacuous... the of_locally_bounded wants ∀ K ⊆ s, IsCompact K → ∃ u, ... — for K = ∅ any u works (the ∀ k ∈ K is vacuous) — but my extraction needs nonempty. Handle: by_cases K = ∅: if empty, take u := 0 family (Summable 0 ✓, vacuous bounds ✓); else obtain nonempty. Let me restructure:\n```lean\n  intro K hK hKc\n  by_cases hKe : K = ∅\n  · subst hKe\n    exact ⟨0, summable_zero, fun m x hx => by simp at hx⟩\n  · obtain ⟨x0, hx0⟩ := Set.nonempty_iff_ne_empty.mpr hKe\n    obtain ⟨c, hcK, hcle⟩ := hKc.exists_isMinOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n    obtain ⟨C, hCK, hCle⟩ := hKc.exists_isMaxOn ⟨x0, hx0⟩ (continuous_id.continuousOn)\n    have hc0 : 0 < c := hK hcK\n    have hCa : 0 ≤ C := hc0.le.trans (hcle C hCK)\n    refine ⟨fun m => C / ((m : ℝ) + 1) ^ 2, summable_div_add_one_sq C, fun m x hx => ?_⟩\n    have hx0 : 0 < x := hK hx\n    have hxc : c ≤ x := hcle x hx\n    have hxC : x ≤ C := hCle x hx\n    rw [Real.norm_eq_abs, abs_of_nonneg (dpsiTermR_nonneg hx0)]\n    calc dpsiTermR x m ≤ x / ((m : ℝ) + 1) ^ 2 := dpsiTermR_le hx0 m\n      _ ≤ C / ((m : ℝ) + 1) ^ 2 := by\n          exact (div_le_div_iff_of_pos_right (by positivity : (0 : ℝ) < ((m : ℝ) + 1) ^ 2)).mpr hxC\n```\nIsMinOn's content: IsMinOn f s c : ∀ y ∈ s, f y ... — actually IsMinOn f s x := ∀ y ∈ s, f x ≤ f y — with f = id: id c ≤ id y — i.e. c ≤ y — `hcle x hx : id c ≤ id x` — simp at it or it's defeq c ≤ x ✓ use directly — hmm IsMinOn def: `IsMinOn f s a := ∀ b ∈ s, f a ≤ f b`? — yes. So hcle x hx : id c ≤ id x — defeq c ≤ x ✓.\n\nHmm wait — IsMinOn signature: `IsMinOn (f : α → β) (s : Set α) (a : α) : Prop := ∀ b ∈ s, f a ≤ f b`? — I think IsMinOn f s a means a minimizes f ON s: ∀ b ∈ s, f a ≤ f b ✓. And exists_isMinOn : IsCompact s → s.Nonempty → ContinuousOn f s → ∃ a ∈ s, IsMinOn f s a ✓ (the extreme value theorem) — signature: `IsCompact.exists_isMinOn (hK : IsCompact s) (hne : s.Nonempty) (hf : ContinuousOn f s) : ∃ a ∈ s, IsMinOn f s a`? — roughly ✓.\n\nAlso the div_le_div_iff_of_pos_right — from session 018's grep: `div_le_div_iff_of_pos_right (hc : 0 < c) : a / c ≤ b / c ↔ a ≤ b` at GroupWithZero/Basic.lean:1149 ✓.\n\n5. **Line 148**: dpsiSeqR_tendstoLocallyUniformlyOn's statement — `TendstoLocallyUniformlyOn dpsiSeqR DinfR atTop (Set.Ioi 0)` — the error: DinfR has type ℝ → ℝ but expected ℕ → ℝ — the argument order of TendstoLocallyUniformlyOn: TendstoLocallyUniformlyOn (F : ι → α → β) (f : α → β) (l : Filter ι) (s : Set α)?? — with F : ι → (α → β) — my dpsiSeqR : ℝ → ℕ → ℝ has the x FIRST — the TendstoLocallyUniformlyOn wants (fun n => fun x => ...) with n : ι FIRST. My dpsiSeqR (x) (n) has x first — MISMATCH. Fix: define dpsiSeqR with n first? But then the deriv usage hasDerivAt (fun y => logGammaSeq y n) (dpsiSeqR ... n)?? hmm — the HasDerivAt's derivative value is dpsiSeqR ? n — the function-of-x form... Let me look at how I use it:\n   - hf' : TendstoLocallyUniformlyOn dpsiSeqR' DinfR atTop (Ioi 0) where the sequence is n ↦ (x ↦ ...) — so define dpsiSeqR (n : ℕ) (x : ℝ) : ℝ := Real.log n − Σ_{m ∈ range(n+1)} 1/(x+m) — n FIRST.\n   - hf : HasDerivAt (fun y => logGammaSeq y n) (dpsiSeqR n y) y ✓ consistent.\n   Let me flip the definition's argument order. Then dpsiSeqR_eq : dpsiSeqR n x = ... and the congr in hsum: hsum.congr (fun n x hx => ...) — the TendstoLocallyUniformlyOn.congr : F = F' eventually-ish → ... — signature: TendstoLocallyUniformlyOn.congr (h : TendstoLocallyUniformlyOn F f l s) (hfg : ∀ n, EqOn-ish...)?? — hmm — actually the issue in the error at 158 (h2 unknown) — inside my hA block I referenced h2 after... let me recheck hA's structure — I had `have h2 : Tendsto ... := by ... rw [h2]` — wait:\n```lean\n    have h2 : Tendsto (fun n : ℕ => Real.log n - (harmonic n : ℝ)) atTop\n        (𝓝 (-Real.eulerMascheroniConstant)) := by\n      have h3 : (fun n : ℕ => Real.log n - (harmonic n : ℝ)) =\n          fun n => -((harmonic n : ℝ) - Real.log n) := by\n        funext n; ring\n      rw [h2]  -- BUG: rw [h2] inside its own proof!\n      exact h.neg\n```\nBUG: `rw [h2]` inside h2's own proof — should be `rw [h3]`. The error at 158 \"unknown identifier h2\" — because within h2's proof, h2 isn't yet available — I wrote rw [h2] — no wait I wrote `rw [h2]` — hmm the code shows `rw [h2]` — should be `rw [h3]`. Let me fix.\n\nAlso the hsum.congr — TendstoLocallyUniformlyOn.congr signature: `TendstoLocallyUniformlyOn.congr (h : TendstoLocallyUniformlyOn F f p s) (hfg : ∀ n, F n =ᶠ[...] F' n)?` hmm — or pointwise: let me check. There are several congr variants. The cleanest: prove the final TendstoLocallyUniformlyOn statement with the dpsiSeqR function directly — i.e. make the DECOMPOSITION the other direction: show dpsiSeqR n x = A + B + C pointwise and rewrite the STATEMENT's function via congr. TendstoLocallyUniformlyOn.congr (hf : TendstoLocallyUniformlyOn F f ...) (h : ∀ n, ∀ x ∈ s, F n x = G n x) : TendstoLocallyUniformlyOn G f ...? — let me check the exact name/form at compile — or restructure: prove TendstoLocallyUniformlyOn (fun n x => dpsiSeqR n x)... — hmm.\n\nActually simpler: make the three-piece assembly the PRIMARY: \n```lean\n  have hdecomp : (fun n => fun x : ℝ => dpsiSeqR n x) =\n      (fun n => fun x : ℝ => (Real.log n - (harmonic n : ℝ)) + (-1 / x + ∑ m ∈ Finset.range n, dpsiTermR x m)) := by\n    funext n\n    funext x\n    exact dpsiSeqR_eq n x\n  rw [hdecomp]  -- in the goal\n  exact (hA.add (hB.add hC))\n```\nwith dpsiSeqR_eq n x : dpsiSeqR n x = (log n − harmonic) + (−1/x + Σ ...) ✓ (flip the args in dpsiSeqR_eq too). Then the goal `TendstoLocallyUniformlyOn dpsiSeqR DinfR atTop (Ioi 0)` — rw [hdecomp]?? — the goal's dpsiSeqR is the FUNCTION `dpsiSeqR` — hdecomp's LHS is the eta-expanded (fun n => fun x => dpsiSeqR n x) — rw [hdecomp] would rewrite occurrences of the eta-form — the goal has bare `dpsiSeqR` — mismatch. Instead: `show TendstoLocallyUniformlyOn (fun n => fun x : ℝ => dpsiSeqR n x) DinfR atTop (Ioi 0)` (eta defeq) then rw [hdecomp] then exact. Hmm — and DinfR vs the decomposed limit: DinfR x = −γ − 1/x + ∑' — the assembly (hA.add (hB.add hC))'s limit function: fun x => (−γ) + ((−1/x) + ∑' dpsiTermR x m) — vs DinfR x := −γ − 1/x + ∑' — the association: −γ − 1/x + Σ vs (−γ) + (−1/x + Σ): −γ − 1/x parses as (−γ) − (1/x) — and (−γ) + (−1/x + Σ) = −γ − 1/x + Σ — ring-equal but not syntactic — so need a final congr on the LIMIT function too: TendstoLocallyUniformlyOn.congr_right? — from derivWithin_tsum's usage: `.congr_right (fun _ hb ↦ (hg.tsum_eqOn hb).symm)` — TendstoLocallyUniformlyOn.congr_right (h) (hfg : EqOn-ish f g)? — let me check its signature — from usage: (....).congr_right (fun t ht => (hg.tsum_eqOn ht).symm) — takes ∀ x ∈ s, f x = g x — so `TendstoLocallyUniformlyOn.congr_right (hf : TendstoLocallyUniformlyOn F f l s) (h : ∀ x ∈ s, g x = f x)?` — direction ambiguous; I'll structure: the final assembly's limit is exactly DinfR:\n```lean\n  have hlimit : (fun x : ℝ => -Real.eulerMascheroniConstant + (-1 / x + ∑' m : ℕ, dpsiTermR x m)) = DinfR := by\n    funext x\n    rw [DinfR]\n    ring\n  rw [hlimit] at hAsum  -- wherever\n```\nLet me now write the corrected Section 2 wholesale (replacing the buggy version). Also fix: dpsiSeqR argument order (n first), dpsiSeqR_eq's signature (n x), the hA/hB/hC blocks, and the final assembly.\n\nAlso `tendstoLocallyUniformlyOn_const` — check name exists: `tendstoLocallyUniformlyOn_const : TendstoLocallyUniformlyOn (fun _ => f) f l s`? — probably `tendstoLocallyUniformlyOn_const (f) (l) (s)`? hmm — there might be `TendstoUniformlyOn.const`? If missing: prove via the same Metric.tendstoUniformlyOn_iff route as hA. Let me include a fallback inline... Actually simplest to verify with a quick grep.","signature":""},{"type":"text","text":"A cascade of small issues. Let me check two names, then rewrite Section 2 wholesale."},{"type":"tool_use","id":"Bash_314","name":"Bash","input":{"command":"cd \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib\" && grep -rn \"tendstoLocallyUniformlyOn_const\\|TendstoLocallyUniformlyOn.congr\\b\\|TendstoLocallyUniformlyOn.congr_right\" Mathlib/Topology/UniformSpace/ --include=\"*.lean\" -r | head -6; grep -rn \"theorem tendstoLocallyUniformlyOn_const\\|lemma tendstoLocallyUniformlyOn_const\" Mathlib/ -r --include=\"*.lean\" | head -2","description":"Check locally-uniform const/congr lemma names"},"extra_content":null}]