[{"type":"thinking","thinking":"Now I understand the error locations:\n\n1. **Line 56** — docstring without declaration. Fix: change to plain comment.\n\n2. **Line 2070** — `ring` after `simp only` closed the goal in marginTilde_eq_sub. Fix: drop `ring`.\n\n3. **Line 2282** — `convert h using 2 <;> field_simp <;> ring` in hD2: field_simp made no progress (×2 — two goals from convert). The convert-produced goals are likely function-equality (closed by rfl via using 2?) and derivative-value equality. h : HasDerivAt (sum-of-pieces function) (sum-of-derivs value) T. Target function: `fun t => c₁*(log t + 1) + 0.137*t⁻¹ + 0.443*((log t)⁻¹*t⁻¹)`; h's function after .add/.const_mul: Pi.add_apply/Pi.mul_apply form — `convert h using 2` on HasDerivAt: congruence down to function and value. The function equality should be closed by rfl (defeq via Pi simp lemmas — but convert doesn't use simp). The VALUE equality: c₁*T⁻¹ + 0.137*(-1/T²) + 0.443*(hu.mul hv derivative)... hu.mul hv derivative value: (-T⁻¹/(log T)^2)*T⁻¹ + (log T)⁻¹*(-1/T^2) — needs field_simp+ring to match -(log T + 1)/((T log T)^2). The error \"field_simp made no progress\" at both goals — meaning convert produced 2 goals where field_simp does nothing (maybe the goals are already closed by rfl? No — then \"no goals\" error). Hmm, the two errors at 2282 + unsolved goals at 2312... \n\nLet me restructure hD2 to avoid convert: follow the file's dominant pattern: compute e : (h's derivative value) = (target value) by field_simp/ring, then `rwa [e] at h` — but h's FUNCTION form also needs to match the target's. In psiOf_log_log_shift etc., the functions built via .mul matched syntactically because I wrote the h2 statement with exactly the built form. Here h's function: ((fun t => (1/(2π)) * (log t + 1)) + ...)? No: hA : HasDerivAt (fun t => (1/(2π)) * (Real.log t + 1))? const_mul: HasDerivAt.const_mul (hf : HasDerivAt f f' x) (c) : HasDerivAt (fun x => c * f x) (c * f') x. So hA's function: fun t => (1/(2π)) * (Real.log t + 1) ✓ matches. hB := hv.const_mul 0.137: fun t => 0.137 * t⁻¹ ✓. hC := (hu.mul hv).const_mul 0.443: hu.mul hv : HasDerivAt (fun t => (Real.log t)⁻¹ * t⁻¹) (...) then const_mul: fun t => 0.443 * ((Real.log t)⁻¹ * t⁻¹) ✓. h := (hA.add hB).add hC: function fun t => (c₁*(log t+1) + 0.137*t⁻¹) + 0.443*(...) — add rule: fun x => f x + g x — so function is fun t => ((1/(2π))*(log t+1) + 0.137*t⁻¹) + 0.443*((log t)⁻¹*t⁻¹). Target: fun t => (1/(2π))*(log t+1) + 0.137*t⁻¹ + 0.443*((log t)⁻¹*t⁻¹) — parses as ((A + B) + C) ✓ SAME. So function sides match syntactically? .add produces `fun x => f x + g x` — HAdd.hAdd applied under lambda vs target's fun t => A + B + C — the target elaborates to fun t => HAdd.hAdd (HAdd.hAdd ...) ... — should match by unification ✓.\n\nThe derivative VALUE: h's value = ((1/(2π))*T⁻¹ + 0.137*(-1/T^2)) + 0.443*((-T⁻¹/(log T)^2)*T⁻¹ + (log T)⁻¹*(-1/T^2)). Target value: (1/(2π))*T⁻¹ + 0.137*(-1/T^2) + 0.443*(-(log T + 1)/((T*log T)^2)). Equal by field_simp+ring. So restructure:\n```\n    have h := (hA.add hB).add hC\n    have hT0' : T ≠ 0 := ne_of_gt hT0\n    have hL1' : Real.log T ≠ 0 := ne_of_gt hL1\n    have e : (1 / (2 * Real.pi)) * T⁻¹ + 0.137 * (-1 / T ^ 2) +\n        0.443 * ((-T⁻¹ / (Real.log T) ^ 2) * T⁻¹ + (Real.log T)⁻¹ * (-1 / T ^ 2)) =\n        (1 / (2 * Real.pi)) * T⁻¹ + 0.137 * (-1 / T ^ 2) +\n          0.443 * (-(Real.log T + 1) / ((T * Real.log T) ^ 2)) := by\n      field_simp [hT0', hL1']\n      ring\n    rwa [e] at h\n```\nBut h's derivative value — what EXACTLY does the elaborated h look like? hA.add hB value: f' + g' where f' = (1/(2π)) * T⁻¹... wait — hA := ((Real.hasDerivAt_log ...).add_const 1).const_mul (1/(2π)): derivative value (1/(2π)) * (T⁻¹)? Real.hasDerivAt_log : HasDerivAt Real.log T⁻¹ T; .add_const 1 : HasDerivAt (fun t => Real.log t + 1) T⁻¹ T (value unchanged); .const_mul c : HasDerivAt (fun t => c * (Real.log t + 1)) (c * T⁻¹) T. So hA's value = (1/(2π)) * T⁻¹ ✓.\n\nhB := hv.const_mul 0.137: value 0.137 * (-1/T^2) ✓.\nhC := (hu.mul hv).const_mul 0.443: hu.mul hv value = (-T⁻¹/(log T)^2) * T⁻¹ + (log T)⁻¹ * (-1/T^2)? mul rule: HasDerivAt.mul (hu : HasDerivAt u u' T) (hv : HasDerivAt v v' T) : HasDerivAt (fun t => u t * v t) (u' * v T + u T * v') T. u = (log ·)⁻¹, u' = -T⁻¹/(log T)^2; v = (·)⁻¹, v' = -1/T^2. Value: u' * v T + u T * v' = (-T⁻¹/(log T)^2) * T⁻¹ + (log T)⁻¹ * (-1/T^2) ✓. hC value: 0.443 * that ✓.\n\nh value: ((1/(2π))*T⁻¹ + 0.137*(-1/T^2)) + 0.443*(...) — note the add order: (hA.add hB).add hC: value = (hA' + hB') + hC' ✓.\n\nTarget value in hD2 statement: (1/(2π))*T⁻¹ + 0.137*(-1/T^2) + 0.443*(-(log T+1)/((T*log T)^2)) — parses ((A)+(B))+(C) ✓.\n\nSo `e` as above: LHS must match h's value syntactically for rwa [e] at h: h : HasDerivAt F value_h T; e : value_h = value_target; rw [e] at h : HasDerivAt F value_target T = goal ✓. The e LHS: (1/(2π))*T⁻¹ + 0.137*(-1/T^2) + 0.443*((-T⁻¹/(Real.log T)^2)*T⁻¹ + (Real.log T)⁻¹*(-1/T^2)). Hmm — mul rule value: is it u' * v T + u T * v' or u T * v' + u' * v T? Mathlib HasDerivAt.mul: `HasDerivAt (fun x => f x * g x) (f' * g x + f x * g')`? Let me check the exact form — in psiOf_log_log_shift's hu.mul hv conversion they wrote:\n```\nhave e : (-(T + 2)⁻¹ / Real.log (T + 2) ^ 2) * (T + 2)⁻¹ + (Real.log (T + 2))⁻¹ * (-1 / (T + 2) ^ 2) = ...\nrwa [e] at h\n```\nSo the form is u' * v T + u T * v' ✓ (f' * g x + f x * g'). Good — matches my e LHS. But wait — .const_mul value: HasDerivAt.const_mul c hf : HasDerivAt (fun x => c * f x) (c * f') x — value c * f' ✓.\n\nOne subtlety: hA's function after add_const: `fun t => Real.log t + 1` then const_mul: `fun t => c * (Real.log t + 1)` ✓ matches target piece `fun t => (1/(2π)) * (Real.log t + 1)` ✓.\n\nSo replace the convert line with the e/rwa pattern. But careful — does the FUNCTION of h match hD2's stated function syntactically for the rwa to close? After rw [e] at h, h : HasDerivAt F_h value_target T where F_h = fun t => ((c₁*(log t+1)) + 0.137*t⁻¹) + 0.443*((log t)⁻¹*t⁻¹)... wait NO: (hA.add hB).add hC — hA.add hB : HasDerivAt (fun t => c₁*(log t+1) + 0.137*t⁻¹)? add rule: HasDerivAt.add (hf : HasDerivAt f f' x) (hg : HasDerivAt g g' x) : HasDerivAt (fun x => f x + g x) (f' + g') x — function fun x => f x + g x where f = fun t => c₁*(log t+1), g = fun t => 0.137*t⁻¹. So the combined function: fun t => (c₁*(log t+1)) + (0.137*t⁻¹)... when Lean elaborates `fun x => f x + g x` with f,g those lambdas: fun x => (fun t => c₁*(log t+1)) x + (fun t => 0.137*t⁻¹) x — beta-reducible to fun x => c₁*(log x+1) + 0.137*x⁻¹ ✓ defeq. Target: fun t => c₁*(log t+1) + 0.137*t⁻¹ + 0.443*((log t)⁻¹*t⁻¹). After rwa [e] at h, goal hD2 statement vs h: defeq up to beta/assoc — rwa uses defeq for the final match (rwa = rw + assumption; assumption checks defeq ✓). The assoc: h's function is fun x => ((c₁*(log x+1)) + (0.137*x⁻¹)) + (0.443*(...)) — target parses + left-assoc ✓ same.\n\n4. **Line 2305 hsum field_simp then ring**: \"field_simp made no progress\" was 2282 only; 2305's field_simp+ring produced \"unsolved goals\" at 2312? No — 2312:67 is the `absurd` line in the dead-code h2 block. h2 is leftover garbage I accidentally left (a weird `(0:ℝ) < ... → False` lemma with an error inside: mul_nonneg_of_nonpos_of_nonpos (by norm_num) hbrB.le — norm_num proves (0.137:ℝ) ≤ 0? NO, 0.137 > 0! I wrote nonsense — h2 was a mistaken leftover; h2' is the correct one. DELETE the h2 block entirely. That's the 2312 error.)\n\nAlso hsum's field_simp [hT0', hL1', hT1.ne'] then ring — the 2282 field_simp errors were both at hD2's convert (two goals). hsum might be OK but let me double check the equation: LHS = D2 − D1/(T−1) with D1, D2 expanded. RHS = c₁(1/T − (log T+1)/(T−1)) + 0.137(−1/T² − 1/(T(T−1))) + 0.443(−(log T+1)/((T log T)²) − 1/(T log T (T−1))). field_simp clears denominators (T, T−1, T², T(T−1), (T log T)², T log T (T−1)) — needs T ≠ 0, T−1 ≠ 0, log T ≠ 0 ✓ provided. Then ring closes. Should be OK. One risk: after rw [hd1, hd2] the expressions contain T⁻¹ (from hd1/hd2's x⁻¹ forms) — field_simp handles ⁻¹ fine.\n\n5. **Line 2328**: heq : N₀ = x (from eq_or_lt_of_le hx with hx : N₀ ≤ x — eq case is N₀ = x). rw [heq] rewrites N₀→x; goal 0 < f x has no N₀ → fails. Fix: `rw [← heq]; exact h0` — ← heq : x = N₀ rewrites x→N₀ ✓.\n\n6. **Line 2331**: DifferentiableOn.continuousOn synthesis — restructure to exact with explicit pieces:\n```\n    have hfc : ContinuousOn f (Set.Icc N₀ x) :=\n      DifferentiableOn.continuousOn (s := Set.Icc N₀ x) (f := f) fun y hy =>\n        (hdiff y (by have : N₀ ≤ y := hy.1; linarith)).differentiableWithinAt\n```\nHmm — DifferentiableOn.continuousOn might take the DifferentiableOn as argument: `DifferentiableOn.continuousOn (h : DifferentiableOn 𝕜 f s) : ContinuousOn f s`. With the lambda given, 𝕜 := ℝ inferred from hdiff ✓. Does it need (𝕜 := ℝ) explicit? The lambda's type DifferentiableWithinAt ℝ f s y determines 𝕜 ✓.\n\n7. **Line 2345**: hconv.mono — need the right name. Search Mathlib for ConvexOn subset lemma: likely `ConvexOn.mono` DOES exist... the error says environment doesn't contain And.mono — dot notation resolved against the STRUCTURE of the type after unfolding? That means ConvexOn is not a structure but a def that reduces to And, and `ConvexOn.mono` doesn't exist in this Mathlib?! Let me grep.\n\n8. **Line 2400**: FAlt_psi_neg {T} implicit — change to explicit (T : ℝ).\n\nAlso fix #1 (line 56 docstring). Let me grep for ConvexOn.mono first.","signature":""},{"type":"text","text":"Now checking whether `ConvexOn.mono` exists in this Mathlib version."},{"type":"tool_use","id":"Bash_58","name":"Bash","input":{"command":"grep -rn \"theorem ConvexOn.mono\\|lemma ConvexOn.mono\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Convex/ | head -3; grep -rn \"ConvexOn.*⊆.*→ ConvexOn\\|ConvexOn.mono\" D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Convex/Function.lean | head -5","description":"Find ConvexOn subset-restriction lemma name"},"extra_content":null}]