[{"type":"thinking","thinking":"Errors:\n1. Line 56: `unexpected token '/-!'; expected 'lemma'` — my replacement comment block: I left the docstring `/-- ... -/` followed by `/-! ## 2. ...` — a docstring `/-- -/` must attach to a declaration. My replacement was ONLY a docstring with no theorem after it — the next thing is `/-! ## 2...` section comment. Fix: make the §1 note a plain comment `/- ... -/` instead of docstring.\n\n2. Line 2070: \"No goals to be solved\" — that's in psiOf_log (field_simp closed the goal, then trailing content?). Line 2070 — let me look. Probably `field_simp [ne_of_gt hT0, hT1]` in psiOf_log closed the goal and there's nothing after... \"No goals\" at 2070:2 — column 2, could be the `ring` after a field_simp somewhere. Let me check line 2070.\n\n3. Line 2282: field_simp made no progress (×2) — in FAlt_psi_neg's hD2 convert...`<;> field_simp <;> ring` — field_simp no progress. The convert h using 2 for hD2 — let me reconsider. Also 2312 unsolved goals.\n\n4. Line 2328: rw [heq] where heq : N₀ = x — eq_or_lt_of_le hx with hx : N₀ ≤ x gives heq : N₀ = x (not x = N₀). So `rw [heq]` rewrites N₀ → x: goal 0 < f x contains no N₀... wait goal is 0 < f x; rw [heq] rewrites occurrences of N₀ with x — goal has no N₀. Fix: `rw [← heq]` rewrites x → N₀: goal 0 < f N₀ = h0 ✓. Since heq : N₀ = x, ← heq : x = N₀ as rewrite replaces x with N₀ ✓.\n\n5. Line 2331: failed to synthesize ContinuousSMul ?𝕜 ℝ — in hfc: `apply DifferentiableOn.continuousOn` — the issue: DifferentiableOn.continuousOn is stated for... the error ContinuousSMul ?𝕜 ℝ suggests the `apply` unified weirdly. Better: build hfc via `(hdiff ...).differentiableWithinAt` — actually DifferentiableOn.continuousOn : DifferentiableOn 𝕜 f s → ContinuousOn f s. Hmm ContinuousSMul ?𝕜 ℝ — the apply can't infer 𝕜? Provide explicitly: `exact differentiableOn_of... ` Alternative: \n```\nhave hfc : ContinuousOn f (Set.Icc N₀ x) := by\n  apply DifferentiableOn.continuousOn\n  intro y hy\n  ...\n```\nThe intro y hy : DifferentiableWithinAt ... — DifferentiableOn f s y def: ∀ y ∈ s, DifferentiableWithinAt ℝ f s y. The `apply DifferentiableOn.continuousOn` leaves goal DifferentiableOn ?f ... — maybe ℝ can't be inferred because ContinuousOn doesn't mention 𝕜. Fix: `exact (DifferentiableOn.continuousOn (𝕜 := ℝ) ?_)`? Hmm DifferentiableOn.continuousOn might have implicit 𝕜. Try: `have hfc : ContinuousOn f (Set.Icc N₀ x) := by apply DifferentiableOn.continuousOn (𝕜 := ℝ)`... Or bypass: ContinuousOn from differentiability — there might be a cleaner path: DifferentiableAt → ContinuousAt: (hdiff y hya).continuousAt, then ContinuousOn via continuousOn_of_forall... Simpler: `ContinuousOn f (Set.Icc N₀ x)` from `DifferentiableOn.continuousOn`:\n```\nexact DifferentiableOn.continuousOn (s := Set.Icc N₀ x) (fun y hy => (hdiff y (by ...)).differentiableWithinAt)\n```\nWith explicit s and the lambda given inline, 𝕜 = ℝ is determined by hdiff's type. Let me restructure hfc/hfd to direct `exact` with explicit fun.\n\n6. Line 2345: `hconv.mono` — ConvexOn is a structure (And) in this Mathlib version and `.mono` isn't a field?? \"Invalid field `mono`: The environment does not contain `And.mono`\" — ConvexOn ℝ s f is DEFINED as Convex ℝ s ∧ (∀ ...)? In recent Mathlib ConvexOn is a def = Convex s ∧ ... no wait — actually the error says hconv has TYPE `Convex ℝ (Set.Ici a) ∧ ∀ ...` — so ConvexOn unfolds to an And and `.mono` resolved against And. That means `ConvexOn.mono` isn't found as dot-notation because ConvexOn is reducible to And?? No — dot notation `hconv.mono` with hconv : ConvexOn ℝ s f should find ConvexOn.mono if ConvexOn is the head... The error indicates the type displayed as the And — ConvexOn must be a def that Lean unfolded for projection... Hmm, actually in Mathlib, `ConvexOn` is defined as:\n```\ndef ConvexOn (s : Set E) (f : E → β) : Prop := Convex 𝕜 s ∧ ∀ ...\n```\nno wait — I recall ConvexOn IS structure-like. The theorem is `ConvexOn.mono : ConvexOn 𝕜 s f → t ⊆ s → ConvexOn 𝕜 t f`? Let me search the actual name in this Mathlib version.\n\n7. Line 2400: FAlt_psi_neg has implicit {T} — marginTilde_convex expects ∀ T, ... — my FAlt_psi_neg is stated `{T : ℝ} (hT : ...) : ...` — implicit T means it IS ∀ T but with implicit binder... the error says \"has type √20+1 ≤ ?m → ...\" vs \"∀ (T : ℝ), ...\" — implicit binders DO match ∀... hmm the error suggests the implicit wasn't generalized: \"has type √20 + 1 ≤ ?m.267 → ...\" — because when using FAlt_psi_neg as a TERM, Lean inserts the implicit as a metavariable: FAlt_psi_neg : {T} → √20+1 ≤ T → ... as a term it's fun {T} hT => ... which IS ∀-shaped... Actually a term with implicit args can be used where ∀ T explicit is expected — Lean inserts mvars and unifies... but here it failed because... the expected type `∀ (T : ℝ), P T` vs provided `@FAlt_psi_neg ?m : √20+1 ≤ ?m → Q ?m` — these don't defeq because ∀ T, P T vs (P ?m →) — eta for implicit... The issue: implicit {T} makes the term's type not literally `∀ T, ...`. Fix: make T explicit in FAlt_psi_neg: `(T : ℝ) (hT : ...)`. But then call sites `FAlt_psi_neg` needs ∀-form: with explicit binders the term IS ∀ T, √20+1 ≤ T → ... ✓. So change {T} to (T) and adjust internal recursive uses — there are none; the block sign lemmas called inside use h1T. Change signature to (T : ℝ) (hT : Real.sqrt 20 + 1 ≤ T).\n\nWait but marginTilde_convex's hψ : ∀ T : ℝ, Real.sqrt 20 + 1 ≤ T → ... — passing `FAlt_psi_neg` with explicit (T)(hT) gives exactly that ∀ ✓.\n\n8. Line 2312:67 unsolved goals — in FAlt_psi_neg hD2's convert/field_simp/ring. And 2282 field_simp no progress ×2 — that's hD2's convert. Let me look at what hD2 produced. Actually 2282 might be the hsum field_simp. Line numbers: let me view the region 2270–2330 to place them precisely.\n\nAlso error at 2070 \"No goals\" — check psiOf_log's field_simp... Let me view lines around these errors.","signature":""},{"type":"text","text":"Multiple fixes needed. Inspecting the error regions."},{"type":"tool_use","id":"Read_56","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","offset":2055,"limit":22},"extra_content":null}]