[{"type":"thinking","thinking":"I wrote a bad syntax in continuous_b2Fun: `(continuous_fun_by_fun_prop := by fun_prop)` — that's garbage. Fix: the argument is `ContinuousOn f I` where I = Icc 0 1. `by fun_prop` proves `Continuous (fun u => (u^2−u)/2)`; then `.continuousOn`. Let me fix that line:\n\n```lean\ntheorem continuous_b2Fun : Continuous b2Fun :=\n  ContinuousOn.comp_fract'' (f := fun u : ℝ => (u ^ 2 - u) / 2)\n    (by fun_prop) (by norm_num)\n```\nHmm — does `by fun_prop` prove ContinuousOn? fun_prop targets Continuous; for ContinuousOn maybe. Safer: `(by continuity : ContinuousOn ℝ (fun u : ℝ => (u^2−u)/2) (Set.Icc 0 1))`? `continuity` proves Continuous... for ContinuousOn goals — continuity handles ContinuousOn too I believe. Let me write `(by continuity)` and adjust.\n\nAlso check comp_fract'' statement once more: `theorem ContinuousOn.comp_fract'' {f : α → β} (h : ContinuousOn f I) (hf : f 0 = f 1) : Continuous fun x => f (fract x)` — with I := Icc 0 1 (local notation \"I\"). The conclusion's function: fun x => f (Int.fract x) — our b2Fun x = (fract x ^2 − fract x)/2 = f(fract x) with f u = (u²−u)/2 ✓ defeq — `exact` should unify (fract x^2 parses as (fract x)^2 ✓).\n\nAlso `HasDerivAt.congr_of_eventuallyEq` — the method direction: `h.congr_of_eventuallyEq hEq` where h : HasDerivAt g ... and hEq : g =ᶠ f? signature: `HasDerivAt.congr_of_eventuallyEq (h : HasDerivAt f f' x) (h₁ : f =ᶠ[𝓝 x] g) : HasDerivAt g f' x`? So h should be HasDerivAt (fun y => y − c) and hEq : (fun y => y−c) =ᶠ fract — but I wrote hEq : fract =ᶠ (fun y => y − c) — direction! Need (fun y => y − ⌊x⌋) =ᶠ fract. My hEq proves fract y = y − ⌊x⌋ — so hEq : fract =ᶠ (fun y => y−c). For congr_of_eventuallyEq I need the OTHER direction: use hEq.symm. Let me fix: `exact h.congr_of_eventuallyEq hEq.symm`.\n\nHmm wait — actually check: `Filter.EventuallyEq.symm : f =ᶠ g → g =ᶠ f` ✓. So `h.congr_of_eventuallyEq hEq.symm`.\n\nAlso `Int.floor_eq_on_Ico` — check name: `Int.floor_eq_on_Ico (n : ℤ) (x : α) (h : x ∈ Set.Ico n (n+1)) : ⌊x⌋ = n`? I believe it exists ✓.\n\nAlso the hf .congr_of_eventuallyEq — might be `Filter.EventuallyEq.hasDerivAt`? — `EventuallyEq.hasDerivAt (h : f =ᶠ[𝓝 x] g) : HasDerivAt f f' x ↔ HasDerivAt g f' x`? or HasDerivAt.congr_of_eventuallyEq ✓ exists with the direction above. Compile will tell.","signature":""},{"type":"text","text":"Fixing the continuity proof syntax error and a congr direction before compiling."},{"type":"tool_use","id":"Edit_191","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean","old_string":"theorem continuous_b2Fun : Continuous b2Fun :=\n  ContinuousOn.comp_fract'' (f := fun u : ℝ => (u ^ 2 - u) / 2)\n    ((continuous_fun_by_fun_prop := by fun_prop).continuousOn) (by norm_num)","new_string":"theorem continuous_b2Fun : Continuous b2Fun :=\n  ContinuousOn.comp_fract'' (f := fun u : ℝ => (u ^ 2 - u) / 2)\n    (by continuity) (by norm_num)"},"extra_content":null}]