[{"type":"thinking","thinking":"Fixes:\n\n1. Line 74: `by continuity` failed (aesop-based?) — the error says aesop failed on the ContinuousOn goal. Use: `(continuous_ofReal ... )` no — simple: `((continuous_fun_name)...)`. For polynomial: `(by fun_prop : Continuous (fun u : ℝ => (u^2 - u)/2)).continuousOn`. fun_prop proves Continuous ✓ then .continuousOn.\n\n2. `Set.Ico_mem_nhds` unknown — it's `Set.Ico_mem_nhds (ha : a < x) (hb : x < b) : Ico a b ∈ 𝓝 x`? The name might be `Set.Ico_mem_nhds` without Set. prefix: `Ico_mem_nhds`. Let me use `Ico_mem_nhds`.\n\n3. The `rw [sub_zero] at h` failed because h : HasDerivAt (fun x_1 => id x_1 − ↑⌊x⌋) 1 x — the derivative is already 1 (sub_const handled it) — and the function has `id x_1 − c` form. So drop the rw. But the congr: h : HasDerivAt (fun x_1 => id x_1 − ↑⌊x⌋) 1 x; goal: HasDerivAt Int.fract 1 x; hEq : Int.fract =ᶠ fun y => y − ↑⌊x⌋ — the function in h is `fun x_1 => id x_1 − ↑⌊x⌋` vs hEq's `fun y => y − ↑⌊x⌋` — id x vs y — defeq ✓ hopefully. `h.congr_of_eventuallyEq hEq.symm` — hEq.symm : (fun y => y − ↑⌊x⌋) =ᶠ Int.fract ✓ — matches h's function modulo id-defeq.\n\n4. Line 92 ring_nf no progress: the convert produced... `convert h2 using 1` — h2 : HasDerivAt (fun x => (fract x^2 − fract x)/2) ((2·fract x·1 − 1)/2)?? The derivative value from ((hf.pow 2).sub hf).div_const 2: hf.pow 2 : HasDerivAt (fract^2) (2·fract x^(2−1)·1) = (2 * fract x ^ 1 * 1)?; .sub hf: (2·fract x^1·1 − 1); .div_const 2: ((2·fract x^1·1 − 1)/2). convert h2 using 1 leaves goal: Int.fract x − 1/2 = (2·(Int.fract x)^1·1 − 1)/2 — ring_nf \"made no progress\"?? The goal is an equality of reals — ring should close: u − 1/2 = (2·u^1·1 − 1)/2 ✓. The error says ring_nf made no progress — maybe the goal after convert wasn't that equality but something with HasDerivAt mismatch... Since convert happened at line 91-92: `convert h2 using 1` then `ring`. Hmm \"ring_nf made no progress\" suggests the remaining goal wasn't a plain equation — maybe the FUNCTION sides differ: b2Fun unfolds to (fract x^2 − fract x)/2 vs h2's function (fun x => (Int.fract x^2 − Int.fract x)/2)? Wait — precedence: `Int.fract x ^ 2` = (Int.fract x)^2 ✓ same. Hmm — the issue: after `unfold b2Fun`, the goal is HasDerivAt (fun x => (Int.fract x ^ 2 − Int.fract x) / 2) (Int.fract x − 1/2) x. h2's function: ((hf.pow 2).sub hf).div_const 2 — that's HasDerivAt ((fun x => fract x^2) − fun x => fract x)/2... the POINTWISE operations: (fun x => fract x ^2 − fract x)/2 — should be the same function up to defeq (Pi sub/div application). convert ... using 1 would produce goals: function equality (defeq, closed by convert's rfl?) and derivative value equality. The error \"ring_nf made no progress\" at 92 — my `ring` was turned into ring_nf? No — I wrote `ring`. The message shows ring_nf tried. In Lean4 Mathlib, `ring` calls ring_nf internally... \"made no progress\" means the goal isn't a ring identity — probably a leftover HasDerivAt-related goal. Let me instead:\n```lean\n  unfold b2Fun\n  have h2 := ((hf.pow 2).sub hf).div_const 2\n  convert h2 using 1\n  · ring\n```\nwith explicit bullet. Or avoid convert: \n```lean\n  have h2 : HasDerivAt (fun x : ℝ => (Int.fract x ^ 2 - Int.fract x) / 2) (Int.fract x - 1/2) x := by\n    convert ((hf.pow 2).sub hf).div_const 2 using 1\n    ring\n  exact h2\n```\nHmm — cleaner: compute the derivative form directly:\n((hf.pow 2).sub hf).div_const 2 : HasDerivAt (((fun x => fract x^2) − (fun x => fract x))/2)?? — HasDerivAt.div_const: (h : HasDerivAt f f' x) : HasDerivAt (fun x => f x / c) (f' / c) ✓ so h2 : HasDerivAt (fun x => (fract x^2 − fract x)/2) ((2*fract x^(2-1)*1 − 1)/2)?? the sub: HasDerivAt.sub (h1 : HasDerivAt f f') (h2 : HasDerivAt g g') : HasDerivAt (fun x => f x − g x) (f' − g') ✓. hf.pow 2 : HasDerivAt (fun x => fract x ^ 2) (2 * fract x ^ (2−1) * 1)?? `HasDerivAt.pow (h : HasDerivAt f f' x) (n) : HasDerivAt (fun x => f x ^ n) (n * f x^(n−1) * f')`? the form: n • ... hmm — `HasDerivAt.pow (hf) (n : ℕ) : HasDerivAt (fun x => (f x)^n) (n * (f x)^(n-1) * f')`? I recall the value is `(n : ℝ) * f x ^ (n - 1) * f'`... something like that. So h2's derivative = (2·(fract x)^1·1 − 1)/2 = (2·fract x − 1)/2 = fract x − 1/2 ✓ ring identity.\n\nSo after `unfold b2Fun; convert h2 using 1` — remaining goal: `Int.fract x - 1/2 = (2 * Int.fract x ^ 1 * 1 - 1) / 2` (or whatever the exact normal form) — ring closes ✓. The failure suggests convert produced a DIFFERENT goal (maybe two goals: function + value, and my single `ring` hit the function one which isn't a ring identity... convert with using 1 on HasDerivAt: congruence on the STRUCTURE: HasDerivAt f₁ v₁ x = HasDerivAt f₂ v₂ x — convert using 1 gives f₁ = f₂ (defeq-closed?) and v₁ = v₂ (the ring identity). If the function goal wasn't closed automatically, `ring` hit `fun x => ... = fun x => ...` — not a ring identity — \"no progress\". Hmm but convert should close defeq ones... The function sides: goal's (after unfold): fun x => (Int.fract x ^ 2 − Int.fract x)/2; h2's: fun x => (Int.fract x ^ 2 − Int.fract x)/2 — IDENTICAL syntactically? hf.pow 2 gives fun x => (Int.fract x)^2 — precedence `(Int.fract x) ^ 2` ✓ same. So function goal closed by rfl in convert; the value goal: `Int.fract x − 1/2 = ?value`. ?value from the chain: HasDerivAt.pow's value form — in current Mathlib: `HasDerivAt.pow (hf : HasDerivAt f f' x) (n : ℕ) : HasDerivAt (fun y => f y ^ n) ((n : _) * f' * f x ^ (n - 1))`?? — the exact form varies by version! `ring` should still close u − 1/2 = (2·1·u^1 − 1)/2 or (2·u^1·1 − 1)/2 etc. — all ring-identical ✓. UNLESS the value has a form like `OfNat.ofNat 2 • ...`?? Hmm.\n\nLet me just try the variants in a quick compile. Actually — simpler robust approach: use `HasDerivAt.congr_deriv`? no. Use the direct chain on the VALUE with simp:\n```lean\n  have h2 := ((hf.pow 2).sub hf).div_const 2\n  convert h2 using 1\n  ring_nf\n```\nIf that fails, inspect the actual goal. Let me just recompile with bullets and see the error details. Actually — the error at 92:2 was on my `ring` — it said \"ring_nf made no progress on the goal\" but didn't show the goal (truncated by head -40 maybe). Let me fix the other two and recompile to see.\n\nAlso fix: drop `rw [sub_zero] at h` (h already has derivative 1 — sub_const's value is 1 − 0? The error showed h : HasDerivAt (fun x_1 => id x_1 − ↑⌊x⌋) 1 x — value already 1 ✓ so sub_zero rw unneeded AND it failed since no `?a − 0` pattern — the value was already 1).","signature":""},{"type":"text","text":"Fixing the four issues: continuity via fun_prop + `.continuousOn`, `Ico_mem_nhds` (no `Set.` prefix), drop the failed `sub_zero` rewrite, and restructure the derivative-value step."},{"type":"tool_use","id":"Edit_194","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    (by continuity) (by norm_num)","new_string":"theorem continuous_b2Fun : Continuous b2Fun :=\n  ContinuousOn.comp_fract'' (f := fun u : ℝ => (u ^ 2 - u) / 2)\n    ((by fun_prop : Continuous (fun u : ℝ => (u ^ 2 - u) / 2)).continuousOn) (by norm_num)"},"extra_content":null}]